AN EXPLICIT VERSION OF CHEBOTAREV’S DENSITY THEOREM SOURABHASHIS DAS Master of Science (5 Year Integrated), National Institute of Science Education and Research, India, 2018 A thesis submitted in partial fulfilment of the requirements for the degree of MASTER OF SCIENCE in MATHEMATICS Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA Sourabhashis Das, 2020 AN EXPLICIT VERSION OF CHEBOTAREV’S DENSITY THEOREM SOURABHASHIS DAS Date of Defence: December 18, 2020 Dr. Habiba Kadiri Associate Professor Ph.D. Thesis Supervisor Dr. Nathan Ng Professor Ph.D. Thesis Supervisor Dr. Joy Morris Professor Ph.D. Thesis Examination Committee Member Dr. Andrew Fiori Assistant Professor Ph.D. Thesis Examination Committee Member Dr. John Sheriff Assistant Professor Ph.D. Chair, Thesis Examination Com- mittee Abstract Chebotarev’s density theorem generalizes the prime number theorem and Dirichlet’s theorem for primes in arithmetic progressions to the setting of number fields. In particular, it asserts that prime ideals are equi-distributed over the conjugacy classes of the Galois group of any given normal extensions of number fields. The first part of the thesis investigates the works by Lagarias and Odlyzko together with the work of Winckler which provides an explicit error term for the prime counting function in Chebotarev’s density theorem. We rework their argument and improve their bounds. The second part improves on the results from the first part by inves- tigating more modern tools. The second part improves further by investigating more modern tools. Some of the main ideas are deriving an explicit formula for a smooth version of a certain prime counting function, and estimating associated sums over the zeros of Hecke L-functions. iii Acknowledgments First and foremost, I have to thank my parents for their love and support throughout my life. Thank you both for giving me the strength to pursue my dreams. My brother deserves my wholehearted thanks as well. I would like to sincerely thank my thesis advisers, Dr. Habiba Kadiri and Dr. Nathan Ng for their guidance and support throughout this study and specially for their confidence in me. Thank you for giving me the freedom to explore the areas I wanted to study during this thesis. I would like to express my gratitude towards Dr. Peng Jie-Wong and Dr. Allysa Lumley for their immense help in understanding some key concepts during the course of this thesis. I would also like to thank Dr. Joy Morris, Dr. Andrew Fiori and Dr. John Sheriff for being members of my supervisory committee, for reading my thesis and giving their thorough feedback on the thesis. To all my friends, thank you for your understanding and encouragement in many, many moments of crisis. Your friendship makes my life a wonderful experience. I cannot list all the names here, but you are always on my mind. Thank you all for being there whenever I needed you. Your friendship has made me a better person. iv Contents Contents v List of Tables viii 1 Introduction 1 1.1 The prime number theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Primes in Chebotarev’s density theorem . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Compilation of ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Studying different versions of Chebotarev’s density theorem [15] and [30] 19 2.1 Artin L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Formula for ψC(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Difference between ψC(x) and IC(x, σ0, T ) . . . . . . . . . . . . . . . . . . 24 2.3 Estimating remainder terms R0(x, T ) and R1(x, T ) . . . . . . . . . . . . . . . . . 28 2.3.1 Bounding S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Bounding S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.3 Bounding S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Reduction to the case of Hecke L-functions . . . . . . . . . . . . . . . . . . . . . 36 2.5 Density of zeros of Hecke L-functions . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6 The contour integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6.1 Bounding Vχ(x, T, U) and Hχ(x, T, U) . . . . . . . . . . . . . . . . . . . . 46 2.6.2 Bounding H∗χ(x, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6.3 Bounding Rχ(x, T, U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 The explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7.1 Estimating Iχ(x, T, U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 v CONTENTS 2.7.2 Explicit bounds depending on T . . . . . . . . . . . . . . . . . . . . . . . 56 2.8 Zero-free regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.8.1 Zero-free region for ζL(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.9 Final Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.9.1 Estimating ψC(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.9.2 Estimating πC(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.10 Reasons for improvements to Winckler’s results and their impact . . . . . . . . . 79 3 A new explicit version of Chebotarev’s density theorem 81 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2 Introducing a smooth weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2.1 Choice of weight g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Introducing smoothed version of ψC(x) . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4 Controlling the smoothed sum over ramified prime ideals . . . . . . . . . . . . . . 89 3.5 Explicit formula for smoothed sum over all prime ideals (IL/K) . . . . . . . . . . 90 3.5.1 Expressing IL/K in terms of Hecke L-functions . . . . . . . . . . . . . . . 90 3.5.2 Obta∣∣ining formula fo∣∣r IL/K . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6 Expressing ∣∣I − |C|L/K |G|xH(1)∣∣ in terms of sum over zeros . . . . . . . . . . . . . . . . 98 3.6.1 Preliminary results about sums over zeros of L(s, χ) . . . . . . . . . . . . 98 3.6.2 Bounding the J (i)’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.7 Study of the sums over the larger zeros . . . . . . . . . . . . . . . . . . . . . . . . 112 3.7.1 Estimating the sum over inverse of zeros S(1)(m,T ) . . . . . . . . . . . . 113 (2) 3.7.2 Estimating SL (m,T, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.7.3 Study of imposter Bessel function K2(zm, wm) . . . . . . . . . . . . . . . 117 3.8 Explicit formula for the error term in the case log x > Xm,T . . . . . . . . . . . . 119 3.8.1 Explicit bounds for Eψ(x) independent of T and δ but dependent on dL . 121 3.8.2 Explicit bounds for Eψ(x) independent of T , δ and dL . . . . . . . . . . . 125 3.9 Explicit bounds for prime ideal counting functions in Chebotarev’s density theo- rem equivalent to θ and π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.10 Comparison to results from Winckler . . . . . . . . . . . . . . . . . . . . . . . . . 129 vi CONTENTS 4 Future work 132 Bibliography 134 vii List of Tables 1.1 Making Lagarias and Odlyzko explicit . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Revising Lagarias and Odlyzko’s and Winckler’s results . . . . . . . . . . . . . . 79 2.2 Improving Winckler’s results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.1 Improvements to α4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 viii Chapter 1 Introduction 1.1 The prime number theorem The main focus of this thesis is to study of distribution of primes in number fields of degree greater than 1. In the base number field, Q, the distribution of primes is studied in the form of the Prime Number Theorem (denoted PNT) and Dirichlet’s Theorem for Primes in Arithmetic Progressions. For x > 1, let us define π(x) = #{p ≤ x | p prime } and the logarithmic integral ∫ x dt Li(x) = . (1.1) 2 log t The famous prime number theorem (PNT) states that π(x) ∼ Li(x), where f(x) ∼ g(x), for positive functions f, g, means lim f(x)x→∞ g(x) = 1. PNT was proven independently by Hadamard and de la Vallée Poussin in 1896. More precisely, they proved √ π(x)− Li(x)  x exp(−c log x), where c is a positive effective constant (effective means the constant c can be computed). The notation f(x)  g(x) means there is a positive constant C such that for all sufficiently large x, |f(x)| ≤ Cg(x). For details of this proof, see Davenport’s book [6, Pages 115 - 124]. 1 1.1. THE PRIME NUMBER THEOREM Let us introduce the classical Riemann ζ-function defined as ∑∞ 1 ∏( )1 −1 ζ(s) = = 1− , ns ps n=1 p for (s) > 1. Here the product is over all primes p. This function is holomorphic in the complex plane except at s = 1 where it has a simple pole. All its zeros outside the so-called critical strip 0 < s < 1 are known and it has been conjectured by Riemann in 1859 that the zeros inside that strip all lie in the line s = 1/2. This conjecture, famously known as the Riemann Hypothesis (denoted RH), remains open to this day. The PNT is equivalent to ψ(x) ∑ l→im∞ = 1 where ψ(x) = log px x pk≤x,k≥1 is a weighted prime counting function. As Riemann showed in his 1859 memoir, the approach of directly using ψ(x) allows us to directly connect to the complex valued Riemann ζ-function and to make use of powerful analytic tools to extract information about prime numbers. We denote the normalized error term in PNT by: ∣∣ − ∣ψ(x) x ∣ E(x) = ∣∣ ∣x ∣. 1 The conjectured size of E(x) is x− +2 and this is equivalent to the infamous Riemann Hypothesis (RH). Unconditional (without RH) results exist (valid for sufficiently large x) and we refer the interested readers to [6, Chapters 17 and 18]. Next, we provide a brief history about the explicit treatment of this error term (so valid for known values of x). This was initiated by wor√ks of√ − log x Rosser and Schoenfeld ([23],[24],[25]) : Rosser in [23, Theorem 22] proved E(x) ≤ log xe 19 for x ≥ e4000 and in [23, Theorem 21] proved E(x) ≤ 0.0119 for x ≥ e50. This was later refined by Dusart [7] in his PhD thesis and successive articles. He proved that E(x) ≤ 9.05× 10−8 for x ≥ e50. Later, Faber (Chinook 2010) and Kadiri [8, Theorem 1.1] proved E(x) ≤ 9.47× 10−10 for x ≥ e50. They introduced a family of smooth weight functions to prove their theorem. This is one of the key tools used in this thesis. This idea was later used by Büthe [5, Theorem 1] (with a different weight). He established bounds on E(x) for smaller values of x and improved 2 1.1. THE PRIME NUMBER THEOREM bounds on E(x) for previously established values of x, for instance, E(x) ≤ 1.12 × 10−10 for x ≥ e50. Finally, Platt and Trudgian proved in 2020: Theorem 1.1. ([20, Theorem 1]). Let R = 5.573412. For each row X,A,B,C, 0 from [20, Table 1], we have that, for all log x ≥ X, ( ) √B ( ) ≤ log x − log xE(x) A exp C , R R and |ψ(x)− x| ≤ 0x. For instance, they obtain (X,A,B,C, 0) = (6000, 611.6, 1.51, 1.94, 4.23 × 10−21) is valid. New work of Fiori, Kadiri and Swindisky (USRA 2020) [9, Theorem 1.2] claims improvements to (X,A,B,C, 0) = (6000, 135.7, 1.5, 2, 1.349× 10−22). Now let a and q be fixed co-prime numbers with q ≥ 3. Let φ denote the Euler phi function defined by φ(q) = #{k ∈ Z | 1 ≤ k ≤ q and gcd(q, k) = 1} and let π(x; a, q) = #{p ≤ x | p ≡ a mod q}. Dirichlet’s prime number theorem for arithmetic progressions states that, for any fixed modulus q, 1 π(x; a, q) ∼ Li(x), as x → ∞. φ(q) This is equivalent to ψ((x; a,)q) ∑l→im∞ = 1, where ψ(x; a, q) = log p.x x k φ(q) p ≤x,k≥1, pk≡a mod q Explicit results regarding the primes in arithmetic progression are well-known. Let ∣∣∣ − x ∣∣ψ(x; a, q) φ(q) ∣E(x; a, q) = x ∣∣. φ(q) In 1984, McCurley generalized the approach of Rosser and Schoenfeld to primes in arithmetic progressions and obtained the first explicit results regarding primes in arithmetic progressions. 3 1.1. THE PRIME NUMBER THEOREM In [17, Theorem 1.2], he obtained bounds for sufficiently large moduli. For instance he proved E(x; a, q) ≤ 11.50 for x ≥ e(log q)2 and q ≥ 1013. In [22, Theorem 1], Ramaré and Rumely proved that E(x; a, q) ≤ 0.002 for x ≥ 10100 and q = 13, and in [14, Theorem 1.1], Kadiri and Lumley using the technique of smooth weights proved that E(x; a, q) ≤ 4.992 · 10−6 for x ≥ 10100 and q = 13. Both results are actually proven for a finite family of “small” moduli (including q = 13) for which the Generalized Riemann Hypothesis (GRH) was partially verified. In order to state precisely GRH, we first introduce Dirichlet characters. A Dirichlet character, χ (mod q) is a completely multiplicative function of period q on integers n ∈ Z which takes complex roots of unity as values when gcd(n, q) = 1 and is 0 otherwise. For a Dirichlet character χ (mod q), the Dirichlet L-function associated to it is given by ∑∞ ∏( )χ(n) − χ(p) −1L(s, χ) = = 1 , ns ps n=1 p for (s) > 1. Here the product is over all primes. χ0 is called the principal character if χ0(n) = 1 for all n with gcd(n, q) = 1 and is 0 otherwise. If we take q = 1, then χ0 is identically 1 and the corresponding L-function is the Riemann ζ-function, ζ(s). It is conjectured that each Dirichlet L-function satisfy the so-called Generalized Riemann Hypothesis, that is that all its zeros in the critical strip align on s = 1/2. It can be proven that they do not vanish on a slim region to the left of s = 1, namely that there exists some absolute positive constant R1 (depending on q) such that L(s, χ) modulo q has at most one zero in the region s > 1− 1R log q and | s| < 1.1 This zero if it exists is real, and is called exceptional (also referred to as Siegel zero). It is another open famous conjecture in analytic number theory that this zero does not exist. Similar to Theorem 1.1, Bennett et al. proved : 2 Theorem 1.2. ([4, Lemma 6.10]). Let R1 = 5.645908801. For q ≥ 105 and x ≥ e4R1(log q) , √ ( √ ) ≤ log x log xE(x; a, q) 1.012xβ0−1 + 1.4579 φ(q) exp − , R1 R1 where the term in β0 is present only if one of the Dirichlet L-function (mod q) possesses an exceptional zero β0. The classical tools to prove above theorems are: 4 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM 1. An explicit formula relating the prime counting functions ψ(x) and ψ(x; a, q) to some sum over the zeros of Riemann ζ-function or Dirichlet L-functions respectively. 2. An estimate of the count for the number of zeros in a specific region of the complex plane. 3. A zero-free region for the corresponding function. 4. Numerical verification of RH / GRH for the first zero. See the latest work of Platt-Trudgian [21] and Platt [19] for current records (for instance if the imaginary part is under 3× 1012, then the zeros of ζ(s) are on the (s) = 1/2 line). All these tools have been adapted to the general number field case recently except the fourth one which is yet to be done. 1.2 Primes in Chebotarev’s density theorem Our next step is to explore prime ideals in number fields. In order to do so, we require some notation and definitions. Let L/K denote a normal extension of number fields with Galois group, Gal(L/K) = G. Let OK denote the ring of integers of the field K. Let N denote the absolute norm of an ideal I in OK (i.e., N(I) = [OK : I]). A prime ideal p in OK is said to be unramified in L if the ideal pOL has a unique decomposition into a product of distinct prime ideals in OL with multiplicity 1. For every unramified prime ideal p in OK , let σp denote the Artin symbol at p. Then Chebotarev’s density theorem essentially tells us that the Artin symbols are equidistributed in the set of conjugacy classes of G. More precisely, Chebotarev proved in 1922 that : Theorem 1.3. (Chebotarev’s density theorem [29]). Let C ⊂ G be a fixed conjugacy class and denote πC(x) = #{p ⊂ OK | p is unramified with N p ≤ x and σp = C}. Then |C| πC(x) ∼ | | Li(x).G This theorem tells us that for a random prime ideal p, the probability that σp equals C is |C| |G| . Note that if K = L = Q, Chebotarev’s density theorem reduces to the Prime Number Theorem (C = G = {1} and πC(x) = π(x)) and if K = Q and L = Q(ζq) where ζq is a primitive q-th root 5 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM of unity, then Chebotarev’s density theorem reduces to Dirichlet’s theorem for primes in a fixed arithmetic progression (modulo q) (C = σa and πC(x) = π(x; a, q)). Let us introduce ζL(s), the Dedekind ζ-function corresponding to the field L, given by the Dirichlet series ∑ 1 ζL(s) = , for (s) > 1, (N(I))s I⊆OL where I runs through the non-zero ideals of the ring of integers, OL. As the Riemann ζ-function, ζL(s) is expected to satisfy the GRH for Dedekind ζ-function : for every complex number s with (s) > 0 and ζL(s) = 0, (s) = 1/2. In 1974, Stark proved the first zero-free region for ζL(s): Theorem 1.4. ([27, lemma 4]). If nL > 1, then ζL(s) has at most one zero ρ = β + iγ in the region |γ| ≤ (4 log d )−1L and β ≥ 1− (4 log dL)−1, (1.2) and if this zero exists, then it has to be real and simple and is denoted as β0. Effective versions of Chebotarev’s density theorem provide explicit error terms which depend on field constants, namely on nL, the degree of extension of L over Q, and on dL, the absolute value of the discriminant of L. These versions are important since many number theoretic applications depend on the size of error terms involved. In this regard, Serre provided explicit bounds which were conditional on the Generalized Riemann Hypothesis (GRH): Theorem 1.5. (Serre [26, Theorem 4]). Let L/K be a normal extension of number fields with G = Gal(L/K). Let C ⊂ G be a conjugacy class. Assume the GRH for the Dedekind zeta function, ζL(s). Then there exists an absolute constant c1 > 0 such that ∣∣∣ | ∣∣ − C|πC(x) | | Li(x)∣∣ ∣ ≤ |C| 1c1 | |x 2 (log dL + nL log(x))G G for all x ≥ 2. In 1977, Lagarias and Odlyzko proved the first unconditional version of Chebotarev Density Theorem: Theorem 1.6. (Lagarias and Odlyzko [15]). Let L/K be a normal extension of number fields with G = Gal(L/K). Let C ⊂ G be a conjugacy class. There exists absolute effectively com- 6 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM putable constants c2 and c3 such that if x ≥ exp(10nL(log dL)2), then ∣∣∣∣ − | | ∣ ∣ | | ( √ )C C π (X) Li(x)∣∣ ≤ Li(xβ log x0C | | | | ) + c2x exp − c3 ,G G nL where β0 term is present only if Dedekind ζ-function, ζL has an exceptional zero β0. Chebotarev’s density theorem is equivalent to lim (ψC(x)) ∑→∞ = 1 where ψC(x) = log(N p).x |C| | |x N p m≤x G p unramified σmp =C Let ∣∣∣ |C|∣ψC(x)− |G|xE (x) = ∣ ∣ ψ |C| ∣∣. (1.3) |G|x Lagarias and Odlyzko proved that: Theorem 1.7. ([15, Theorem 9.2]) There exists c4, c5 > 0 constants such that if x ≥ exp(4nL(log d )2L ), then ( √ ) xβ0−1≤ log xEψ(x) + c4 exp − c5 , β0 nL where β0 term is present only if the Dedekind ζ-function ζL has an exceptional zero β0. In his Ph.D (2013), Winckler made every result in the article of Lagarias and Odlyzko [15] explicit, and he proved: Theorem 1.8. ([30, Theorem 1.2]). Let L/K be a normal extension of number fields with 44/5 G = Gal(L/K). Let C ⊂ G be a conjugacy class. If x ≥ exp(8nL(log(150 867dL ))2), then ∣∣∣ | | ∣∣ − C ∣ ( √ ) πC(X) | | Li(x) ∣ G ∣ ≤ |C| β 1 log x0| | Li(x ) + c6x exp − ,G 99 nL where c6 = 783 846 699 796 966 < 7.84 × 1014 and the β0 term is present only if the Dedekind ζ-function, ζL has an exceptional zero β0. This follows by partial summation and integration by parts from: 7 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM Theorem 1.9. ([30, Theorem 8.2]). Let L/K be a normal extension of number fields with 44/5 G = Gal(L/K). Let C ⊂ G be a conjugacy class. If x ≥ exp(4nL(log(150 867dL ))2), ( √ √ ) xβ0−1≤ 7− 4 3 log xEψ(x) + c7 exp − , β0 5 nL where c7 < 1.51 × 1012 and the β0 term is present only if the Dedekind ζ-function, ζL has an √ exceptional zero β0. Note that 7−4 3 5 = 0.01436 . . .. In this thesis, we will only look at the unconditional versions of Chebotarev’s density theorem. In Chapter 2 of the thesis, we study Winckler’s [30] closely. We were able on one hand to correct some errors and on the other hand to improve some of his results which leads to the improvement of the constants c6 ad c7. For instance, we modify Theorem 1.8 and Theorem 1.9 into the following three theorems: Theorem 1.10. Let L/K be a normal extension o(f number fields with G =)Gal(L/K). Let 44 C ⊂ G be a conjugacy class. If nL ≥ 2 and x ≥ exp 8nL(log(1 114 759 d 5 ))2L , then ∣∣∣ ∣∣ − |C| ∣∣∣ |C| β | ( √ ) C| 1 log x π (x) Li(x) < Li(x 0 ′C |G| |G| ) + |G|c6x exp − , (1.4)99 nL where c′6 = 0.4958 and the β0 term is present only if the Dedekind ζ-function, ζL has an excep- tional zero β0. Theorem 1.11. Let L/K be a normal extension o(f number fields with G =)Gal(L/K). Let 44 C ⊂ G be a conjugacy class. If nL ≥ 2 and x ≥ exp 8nL(log(10 478 733 d 5L ))2 , then ∣∣∣ | | ∣∣∣ | | | | ( √ )∣ − C C C 1 log xπC(x) | Li(x) < Li(xβ0) + c′′ x exp − ,G| ∣ |G| 6 |G| 99 nL where c′′6 = 0.4 and the β0 term is present only if the Dedekind ζ-function, ζL has an exceptional zero β0. Theorem 1.12. Let L/K be a normal extension of number fields with G = Gal(L/K). Let 8 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM ( ) 44 C ⊂ G be a conjugacy class. If n 5 2L ≥ 2 and x ≥ exp 4nL(log(1 114 759 dL )) , then − ( √ √ )xβ0 1 Eψ(x) ≤ + c′7 exp − 7− 4 3 log x , β0 5 nL where c′7 = 5805.17 and the β0 term is present only if the Dedekind ζ-function, ζL has an exceptional zero β0. In Theorem 1.10 and Theorem 1.11, we replace c by c′6 6 = 0.4958 and c′′6 = 0.4 which is an improvement by a factor of 1015. The larger x is, the smaller c6, c ′ 6, c ′′ 6, c7 and c ′ 7 will become. Moreover, we correct errors in Winckler’s work and as a consequence the value 150 867 in Theorem 1.8 should be corrected to 1 114 759 (the same constant that appears in Theorem 1.10). Note that Theorem 1.10 and Theorem 1.11 differ by the choice for one of the key parameters (labelled T0) which occurs in the proofs. More precisely, in Theorem 1.10, we take T0 = 2 and in Theorem 1.11, we take T0 = 44. The reasons behind our improvements to Winckler’s result, although we followed the same techniques he employed, are explained in Section 2.10. We summarize corrections to Winckler’s [30] article in Table 2.1, and improvements in Table 2.2. In Chapter 3, we prove a new explicit version of Lagarias and Odlyzko’s result on Cheb- otarev’s density theorem. Instead of establishing an explicit formula relating ψC(x) to the zeros of Dedekind ζ-function, ζL, we introduce a smooth weight in the definition of ψC(x) and study the approximation ψ̃C(x). We appeal to the theory of inverse Mellin transform instead of using a Perron’s formula (as used in Lagarias and Odlyzko [15] and Winckler [30]) and prove an original explicit formula for ψ̃C(x). We then proceed to study new weighted sums over the zeros of the Dedekind ζ-function. The sums over the zeros can be estimated by using bounds for the number of zeros in a box. We modify the original approach for the sums over the zeros closer to the real line and for the sums over the zeros of the large imaginary part. This leads to a better optimization of one of the key parameters and allows us to establish a result valid for more values of x than in Lagarias and Odlyzko [15] (and consequently of Winckler). For instance, we prove a new result for the number of non-trivial zeros of the Artin L-function, L(s, χ, L/E), where L/E is an abelian extension and χ is a character of Gal(L/E), and thus improve [30, Lemma 5.4]. Note that by class field theory, L(s, χ, L/E) is a Hecke L-function and therefore is 9 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM holomorphic in the case χ = 1. Proposition 1.13. Let 0 < ≤ 1 and a > 0. Let A(χ) and δ(χ) be the non-zero numbers defined in (2.43) and (2.44) respectively. Let T be a real number and let nχ,a,(T ) denote the number of non-trivial zeros ρ = β + iγ of L(s, χ, L/E) with |T − γ| ≤ a. Then nχ,a,(T ) ≤ nE(c1(a, ) log(2 + + |T |) + c2(a, )) + c1(a, ) logA(χ) + 4c1(a, )δ(χ), where (1 + )2 + a2 c1(a, ) = (1.5) 2 and ( )( ) (1 + )2 + a2 1 164 c2(a, ) = + . (1.6) 14 As a consequence, we obtain two theorems. The first one has Eψ(x) dependent on both nL and dL: Theorem 1.14. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let 1/n R = 29.57. Let m ≥ 2 be an integer. If log x ≥ 4mRn (log 88d L)2L L , then xβ0−1 Eψ(x) ≤ + 1(m,x, nL, dL), β0 with { } ( 1 √ ) − 1 m 1 − 2m 2m+1 log x 1(m,x, nL, dL) = λ(m)max (log dL)nL , Dm+1 (log x)m+1 exp (1.7)m+ 1 RnL where λ is defined in (3.163) and the β0 term is present only if the Dedekind ζ-function, ζL has an exceptional zero β0. For the second one, we remove the dependence of dL in Eψ(x): Theorem 1.15. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let 1/n R = 29.57. Let m ≥ 2 be an integer. If log x ≥ 4mRnL(log 88d L)2L , then xβ0−1 Eψ(x) ≤ + 2(m,x, nL), β0 10 1.2. PRIMES IN CHEBOTAREV’S DENSITY THEOREM with ( 1 √ ) 1− 1 1 m+1 − 1.5m 2 log x 2(m,x, nL) = ν(m) nL (log x)m+1 exp (1.8)m+ 1 RnL where ν is defined in (3.167) and the β0 term is present only if the Dedekind ζ-function, ζL has an exceptional zero β0. As a corollary to Theorem 1.15 (taking m = 2), we have Corollary 1.16. Under the assumptions in Theorem 1.15, we have xβ0− ( √ ) 1 2 ≤ 1 − log x ≥ (log d ) 2 L Eψ(x) +A n 3 1 L(log x) 3 exp 0.13 for all log x 19 810 ,β0 nL nL where A1 = 0.0396 if β0 exists and 0.0249 otherwise. We then deduce an explicit bound for πC(x) from ψC(x): Theorem 1.17. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let β0 be the possible exceptional real zero of ζL(s). Then ∣∣∣ | | ∣∣∣ | ( √ )∣ − C ∣ ≤ C| |C| log xπC(x) Li(x) Li(xβ0| | | | ) + E0 | |nLx exp − 0.0919 ,G G G nL for all ≥ (log dL) 2 log x 39 620 nL where E0 = 4.714× 10−6 if β −60 exists and 2.97× 10 otherwise. Corollary 1.18. Under the assumptions in Theorem 1.17, we have ∣∣∣ ∣∣ |C| ∣ | | | | ( √ ) C C log x πC(x)− | | Li(x) ∣∣ ≤ | | Li(xβ0) + E1G G |G|x exp − F1 nL for all log x ≥ G1nL(log d 2L) where E1 = 1.65 × 10−5, F1 = 0.09 and G1 = 9906. Another admissible value for (E1, F1, G1) is (1.23× 10−9, 1/99, 9 906). 11 1.3. COMPILATION OF IDEAS Comparison: Notice that Winckler’s result as shown in Theorem 1.8 gives (E1, F1, G1) = (7.84 × 1014, 1/99, 3 090 ). Our corrections leads to (E1, F1, G1) = (0.4, 1/99, 16 006) as shown in Theorem 1.11 and a new result for Corollary 1.18 gives (1.23× 10−9, 1/99, 9 906). Therefore, we improve both the error term (factor of 1024) and the range (factor of 1.6). Notice that the formula for range of log x given in Theorem 1.17 has nL in the denominator whereas both Theorem 1.11 and Theorem 1.8 have nL in the numerator. Therefore, as the degree of the field increases, our Theorem 1.17 improves the range of log x by a factor of n2L. A more detailed comparison with Winckler’s result is given in Section 3.10. Finally Table 1.1 lists best known explicit versions to Lagarias and Odlyzko’s [15] lemmas used to establish their main theorem. Table 1.1: Making Lagarias and Odlyzko explicit Results from Lagarias and Odlyzko [15] Best explicit version of their results Lemma 5.2 [30, Lemma 4.3] Lemma 5.3 [30, Lemma 4.5] Lemma 5.4 Proposition 1.13 Lemma 5.5 Lemma 3.21 Lemma 5.6 Lemma 3.20 Lemma 6.1 [30, Lemma 4.4] Lemma 6.2 Lemma 2.19 Lemma 6.3 Lemma 2.23 Lemma 8.1 Lemma 2.32 Lemma 8.2 [2, Theorem 1] Theorem 9.2 Theorem 1.15 1.3 Compilation of ideas Next, we mention the tools, ideas and motivations which form the basis of chapter 3. First of all, the idea for using smooth weight has been derived from the works of Faber and Kadiri in [8] who used smooth approximation of the prime counting functions in their work. In chapter 3, we introduce a smooth weight h as in (3.10) approximating the identity on [0, x] and obtain a smooth version of ψC(x), denoted as ψ̃C(x) as in (3.21). We introduce the smooth sum over all primes ideal, IL/K(x) as in (3.26) and relate it to ψ̃C(x) as in (3.25). Then using the inverse Mellin transform of h, we express IL/K(x) as a sum over Artin 12 1.3. COMPILATION OF IDEAS L-functions as in (3.5.1). Since the holomorphicity of the Artin L-functions is yet to be proved, we use Deuring’s reduction to express IL/K(X) as a sum over Hecke L-functions over an intermediate field (which are holomorphic) as in (3.5.1). This allows us to use Cauchy’s residue theorem to obtain an explicit formula for the sum IL/K(x) which involves sums over the zeros of the Dedekind ζ-function, ζL(s) as shown in Proposition 3.16. Next we split this sum over zeros into four different regions as in (3.63), (3.64), (3.65), (3.66) and bound each one separately. This approach is different from Lagarias and Odlyzko’s approach in [15] where they divided the sum over the zeros into three different sums. In particular, we add an extra region to better estimate the sum over the low-lying zeros (zeros close to real line). When bounding the sum over the zeros of ζL(s), two of the most important tools we require are : 1. Zero-free regions (regions which can not contain a single zero of ζL(s)). Inspiration for this comes from the works of Kadiri [13] on explicit zero-free regions for Riemann ζ-function generalized by Ahn and Kwon [2][3] to Dedekind ζ-function. 2. Zero-density formulas (estimates for the number of zeros in a particular region). Recently, Hasanalizade et al. in [11] improved the bound given by Trudgian [28]. We use their result in this thesis as Theorem 3.10. Moreover, since there is no verification of GRH for number fields, a more cautious approach is taken to bound the sums over the low lying zeros of the Hecke L-functions. We prove a new result for the number of non-trivial zeros ρ = β + iγ of the Hecke L-function, L(s, χ) with |T −γ| ≤ a denoted as nχ,a,(T ) where T is any real number and is a constant taken close to 0. This is shown in Proposition 1.13 in this thesis. Note that nχ,a,(T ) generalizes nχ(T ) as in [15, Lemma 5.4] which counts the number of non-trivial zeros ρ = β + iγ of L(s, χ) with |T − γ| ≤ 1. Following this, we also prove a new bound for the logarithmic derivative of L(s, χ) in a restricted range as given in Lemma 3.20. As mentioned earlier, to bound the sum over the non-trivial zeros above an arbitrary height T ≥ 44, we use the zero density formula as proved by Hasanalizade et al. in [11] 13 1.4. NOTATION and additionally generalize the techniques used by Fiorelli and Martin in [10] for Dirichlet L-functions. This results in obtaining integrals which are called Bessel integrals and are given in (3.129) and (3.130). To compute the required bounds for the remaining sum over the non-trivial zeros of the Dedekind ζ-functions, we use types of incomplete Bessel functions as already introduced by Rosser and Schoenfeld [24], and recently studied in more detail by Kadiri and Lumley in [14] and by Bennett et al. in [4]. Finally, we choose an appropriate T as in (3.148) to obtain the error term Eψ as shown in Theorem 1.14 and Theorem 1.15. Our choice is different from Lagarias and Odlyzko [15] and from Winckler [30]. This allows an improvement on the range for log x for both papers. Then, using partial summation and integration by parts, we obtain Theorem 1.17 for πC(x). 1.4 Notation This section introduces notation and definitions which are used in this thesis. Let N, Z, R, C be the set of natural numbers, integers, real numbers and complex numbers respectively. Let dF denote the absolute value of the discriminant of the number field F . Let nF denote the degree of extension of F over Q, [F : Q]. Let OF be the ring of integers of the number field F . Let N denote the absolute norm of an ideal I in OF (i.e., N(I) = [OF : I]). Let ζF (s) denote the Dedekind ζ-function corresponding to the field F and is defined for complex numbers s with real part (s) > 1 by the Dirichlet series ∑ 1 ζF (s) = , ⊆O (N(I)) s I F where I denotes the non-zero ideals of the ring of integers, OF . 14 1.4. NOTATION Let L/K be a normal extension of number fields. Let G denote the Galois group, Gal(L/K) and let C denote a conjugacy class of G. Let ρ : G → GLn(C) be a group representation. Let p denote a prime ideal in OK . Let q be a prime ideal in OL such that q lies over p. Let σp denote the Artin symbol at p. Let L(s, ρ, L/K) denote the Artin L-function attached to ρ. Let φ denote the character of ρ and given by φ = tr ρ. We also write L(s, φ, L/K) to denote L(s, ρ, L/K). Fixing L and K, we denote L(s, φ) for L(s, φ, L/K). Let C be a conjugacy class of G = Gal(L/K), g ∈ C, G0 =< g > be the cyclic group generated by g, E be the fixed field of G0, and χ denote any irreducible character of G0. Let L(s, χ, L/E) or L(s, χ) denote the Hecke L-function related to the character χ. We say f is big oh of g and denote it as f(x) = O(g(x)) or f(x)  g(x) if there is a positive real number c such that for all sufficiently large x, |f(x)| ≤ cg(x). The prime ideal counting functions studied in this thesis are: ∑ ∑ πC(x) = 1, ψC(x) = log(N p) p unramified p unramified N p≤∑x, σ =C N pm≤x, σmp p =∑C θC(x) = log(N p) and θ0(x) = log(N p). p unramified p unramified N p≤x, σp=C N p≤x Let h be a chosen smooth weight with its corresponding Mellin transform H defined as ∫ ∞ H(s) = h(t)ts−1 dt. 0 Let θ characterize the Artin symbol at p coinciding with the conjugacy class C. More 15 1.4. NOTATION specifically, for p unramified in L, we have ⎧⎪⎪⎨ 1 if σmp = C, θ(pm) = ⎩⎪⎪ 0 otherwise, and |θ(pm)| ≤ 1 if p ramifies in L. The smooth version of the prime ideal counting functions studied in the thesis are: ∑ ∑ ( )N pm ψ̃C(x) = (log(N p))h x p unramified m≥1 ∑σmp =∑C ( ) m N p m IL/K(x) = θ(p )(log(N p))h ≥ xp∑m 1 ∑ ( )N pm ĨL/K(x) = θ(p m)(log(N p))h . x p ramifiedm≥1 Let Eψ denote the error term defined as ∣∣∣ ∣∣ψC(x)− |C| |G|x E (x) = ∣∣ψ |C| ∣, |G|x and its smooth version, Eψ̃ is defined as ∣∣∣ ∣∣ ψ̃ |C| C(x)− |G|x ∣ Eψ̃(x) = ∣ | .C| ∣ |G|x The following are related to the Dedekind ζ-function, ζL(s) and the Hecke L-function, L(s, χ): For β, γ ∈ R, ρ = β + iγ denote the zeros of ζL(s) or L(s, χ). Let β0 denote the possible real exceptional zero of ζL(s). Z(ζ) denotes the set of non-trivial zeros of ζL(s), i.e., Z(ζ) = {ρ = β + iγ | ζL(ρ) = 0, 0 < β < 1}. 16 1.4. NOTATION Z(χ) denotes the set of non-trivial zeros of L(s, χ), i.e., Z(χ) = {ρ = β + iγ | L(ρ, χ) = 0, 0 < β < 1}. The zero counting functions used in the thesis are : For t ∈ R and a > 0, nχ(t) = #{ρ = β + iγ | L(ρ, χ) = 0, 0 < β < 1, |γ − t| ≤ 1} and nχ,a,(t) = #{ρ = β + iγ | L(ρ, χ) = 0, 0 < β < 1, |γ − t| ≤ a}, where is some real number taken close to 0 and for T ≥ 1, NL(T ) = #{ρ = β + iγ | ζL(ρ) = 0, 0 < β < 1, |γ| ≤ T}. We define the sums over the zeros, ρ = β + iγ of the Dedekind ζ-function, ζL(s) as: Let T ≥ 1 and α4, RL, D > 0, then ∑ ∣∣ ∣(3) ∣∣ ρ 1 ∣J (x) = x H(ρ)− ∣∣,ρ ρ= 1−β0,|ρ 1∑|< 2 J (4)(x) = xβ−1|H(ρ)|, ρ =β0,|ρ|≥ 1∑,|γ|≤ 1 2 α4 log dL J (5)(x, T ) = xβ−1|H(ρ)|, |ρ∑|≥ 1 , 1 <|γ| 0 and T ≥ 1, we denote ⎪⎪⎧⎨ T √ if log x ≤ X⎪ ( ) m,T , Xm,T = (m+ 1)RL log 2(DT ), and T1 = ⎪⎩W = 1 exp log xD R (m+1) if log x > Xm,T .L For α1, α2, α3, D > 0, we define the function Q as ( ) ( ) ( √ ) nLu Du − nLt Dt D utQ(t, u) = log log + 2α1nL log + 2α2nL + 2α3. π 4πe π 4πe 2 Given positive real numbers n,m, α, β and l, we define an incomplete modified Bessel function of the first kind as ∫ ∞ ( )(log βu)n−1 In,m(α, β; l) = exp − α du. um+1l log βu Given positive constants n, z, and y, we consider a variant of incomplete Bessel function of the second kind as ∫ ( ( )) 1 ∞ K (z; y) = vn−1 z 1 n exp − v + dv. 2 y 2 v For x > 0, the complimentary error function, erfc is given by ∫ 2 ∞ 2 erfc(u) = √ e−t dt. π u 18 Chapter 2 Studying different versions of Chebotarev’s density theorem [15] and [30] In this chapter, we give a survey of articles [15] and [30] and improve the results in [30]. As a consequence, we derive an asymptotic formula with an explicit error term for a weighted prime power counting function ∑ ψC(x) = ψC(x, L/K) = log(N p), N pm≤x p unramified σmp =C and prove the following theorem : Theorem 1.12. Let β0 be the possible exceptional real zero of ζL((s), and χ0 be the charact)er 44 (real) such that the L-function L(β0, χ0) = 0. If nL ≥ 2 and x ≥ exp 4nL(log(1 114 759 d 5 ))2L , then ∣∣∣∣ − |C| |C| ∣ xβ0 ∣ |C| ψC(x) ∣| |x+ | χ0(g) ≤ ψ(x), (2.1)( G G)| β ∣ 0 |G| √ √ where ψ(x) = 5805.17x exp − 7−4 3 log x5 n and the third term in the left hand side of (2.1)L can be suppressed in the absence of the exceptional zero β0. The L-function mentioned in this theorem are introduced and studied in Section 2.1. In general, this chapter can be summarized as the study of the five following steps which will be explained in detail: 1. ψC(x) differs from a truncated inverse Mellin transform IC(x, σ0, T ) defined in (2.8) by a remainder term R1(x, σ0, T ) defined in 2.11 which is shown in Lemma 2.3. Further R1(x, σ0, T ) is bounded in Section 2.3. 19 2.1. ARTIN L-FUNCTIONS 2. IC(x, σ0, T ) can in fact be reduced to a linear combination of logarithmic derivatives of Hecke (abelian) L-functions using During reduction as shown in Section 2.4 and (2.42). 3. IC(x, σ0, T ) is expressed as a sum of Iχ(x, T )’s defined in (2.60) which differs from a certain contour integral Iχ(x, T, U) defined in (2.61) by a remainder term Rχ(x, T, U) which is shown in Section 2.6. This step is traditionally labelled “ shifting the line of integration to the left ”. Certain results on the density of zeros of ζL(s) in the critical strip 0 < Re(s) < 1 are required to estimate Rχ(x, T, U). 4. The contour integral Iχ(x, T, U) is evaluated by Cauchy’s residue theorem. The integrand has poles at the zeros and the poles of ζL(s), and the result is a main term |C| |G|x coming from the pole of ζL(s) at s = 1, together with a certain sum S(x, T ) over the zeros of ζL(s) within the contour BT,U . The end result of these steps is a truncated explicit formula for ψC(x) with an unconditional error term, which is stated as Theorem 2.29 and Theorem 2.30 depending on the position of T . This step is explained in detail in Section 2.7. 5. Finally Section 2.9 gives the required explicit estimates. The asymptotic formula ψC(x) ∼ |C| |G|x with an explicit remainder term is derived by making an appropriate choice of T as a function of x, to minimize the accumulated error terms as shown in Theorem 1.12. The asymptotic formula πC(x) ∼ |C||G| Li(x) with an explicit remainder term is derived by partial summation from that for ψC(x) as shown in Theorem 1.10. Before commencing with the proof, we introduce Artin L-functions. 2.1 Artin L-functions In this section, we give a definition for Artin L-function. This requires a lot of notation. Let L/K be a normal extension of number fields with the Galois group, Gal(L/K) = G. Let ρ : G → GLn(C) be a group representation. Attached to this representation is a meromorphic L-function originally defined by Artin. We shall define this function, giving all required details. Let dL and dK denote the absolute values of the discriminants of L and K, respectively, and let nL = [L : Q] and nK = [K : Q]. Let OL and OK be the ring of integers of the number fields L and K respectively. Let p be a prime ideal in OK . Let q be a prime ideal in OL such that q 20 2.1. ARTIN L-FUNCTIONS lies over p (denoted as q|p). Define the decomposition group as Dq = {σ ∈ G | σ(q) = q}. We know that OK/p can be identified with a finite field Fq for some q = pm where p is prime and m is a positive integer. Furthermore, if [OL/q : OK/p] = f , then we can think of OK/q as Fqf . Therefore we see that Gal(OL/q/OK/p) is isomorphic to Gal(Fqf /Fq). By Galois theory, we get that the group Gal(Fqf /Fq) is cyclic of order f and is generated by the element τ : x → xqq for x ∈ Fqf . Using the isomorphism of groups, we may assume τq to be an element of the group Gal(OL/q/OK/p). We now check that there exists a canonical map from Dq to Gal(OL/q/OK/p) which is defined by sending σ → σ where σ(x + q) = σ(x) + q. We define the inertia group to be Iq = ker(Dq → Gal(OL/q/OK/p)). Iq can also be described as Iq = {σ ∈ G | σ(x) ≡ x (mod q), ∀x ∈ OL}. Therefore, we obtain a canonical isomorphism D /I ∼q q = Gal(OL/q/OK/p). Therefore, we can now choose an element σq ∈ Dq/Iq whose image in Gal(OL/q/OK/p) is the generator τq as described above. Such an element σq is called a Frobenius automorphism at q and it is only well-defined modulo Iq. Notice that σq is no longer an element of the Galois group G but a coset of Iq. Now if p is unramified, then one can show that Iq is a trivial group for every q|p. Besides, since there are finitely many ramified prime ideals in OK , one can deduce that all but finitely many Iq are trivial for q|p. Also, for unramified primes p, we can show that q ranges over the prime ideals above p and the σq’s form a conjugacy class. This class is called the Artin symbol at p, denoted as σp and explicitly defined as σp = {σq | q divides p}. 21 2.2. FORMULA FOR ψC(X) Using the above theory and representations of finite groups, Artin introduced his L-functions that generalize Dirichlet L-function as follows. The Artin L-function attached to ρ is defined by ∏ L(s, ρ, L/K) = Lp(s, ρ, L/K), p where the local L-function at p is defined as ( )−1 L V Iq −s p(s, ρ, L/K) = det I − ρ| (σq)N p for Re(s) > 1. Here, the product runs over prime ideals in OK , N denotes the norm of a non-zero ideal, q denotes a prime ideal above p, V Iq = {v ∈ V | ρ(g)v = v for all g ∈ Iq}, and for our case V = C. Sometimes, we also write L(s, φ, L/K) for L(s, ρ, L/K), where φ = tr ρ denotes the character of ρ. Also, once we fix the fields K and L, we abbreviate L(s, ρ, L/K) to L(s, ρ). One can easily show that L(s, φ1 + φ2) = L(s, φ1) L(s, φ2), for any characters φ1 and φ2 of G. 2.2 Formula for ψC(x) In this section, we will derive an explicit formula for ψC(x). The proof follows the classical arguments for ψ(x) as outlined in Davenport’s book [6, Chapter 17]. Let φ be a irreducible character of G = Gal(L/K). Let us define m 1 ∑ φK(p ) = | | φ(τ mα), (2.2) I α∈I where I is the inertia group of q, one of the prime ideal factors of p, and τ is one of the Frobenius automorphism corresponding to p. If L(s, φ, L/K) is the Artin L-series associated to φ, then from [18, Proposition 2.3.1], we get that for Re(s) > 1, ∑ φ mK(p ) logL(s, φ, L/K) = . (2.3) m(N p)ms p,m 22 2.2. FORMULA FOR ψC(X) Hence taking the derivatives on both side with respect to s, we get L′ ∑− log(N p)(s, φ, L/K) = φ (pmK ) , L (N p)ms p,m where the outer sum is over all the prime ideals of K. To single out those pm with σmp = C, we will use the characters φ. This is done as the following:- Suppose that g ∈ C. We define a function fC : G → C by ∑ fC(h) = φ(g)φ(h), (2.4) φ where φ is the complex conjugate of φ. By the orthogonality relation of characters, we get ⎪⎨⎪ ⎧ |CG(h)| if h ∈ C, fC(h) = ⎪⎪⎩ (2.5)0 otherwise, where |CG(h)| denotes the centralizer of h in G. Hence if −|C| ∑ L′ FC(s) = | | φ(g) (s, φ, L/K), (2.6)G L φ then for Re(s) > 1, we have the Dirichlet series expansion −|C| ∑ L′ |C| ∑ ∑ FC(s) = | | φ(g) (s, φ, L/K) = m −ms G L | | φ(g) φK(p ) log(N p)(N p)G ∑φ ∑ ∑ φ p,m|C| 1 = | | φ(g) m | | φ(τ α) log(N p)(N p) −ms G I ∑(φ ∑p,m α∈I| | )C = φ(g)φ(τm| || | α) log(N p)(N p) −ms. I G p,m φ ∑ α∈I = θ(pm) log(N p)(N p)−ms, (2.7) p,m 23 2.2. FORMULA FOR ψC(X) where for p unramified in L, we have ⎪⎧ m |C| ∑ ⎪⎨ 1 if σm = C,θ(p ) = | || | φ(g)φ(τmα) =I G ⎩⎪⎪ p φ ∈ 0 otherwise,α I and |∑θ(p m)| ≤ 1 if p ramifies in L. Notice that if p unramified in L and σmp = C, then I = {1} and φ(g)φ(τm) = |C (τm)| = |G|φ G |C| which yields θ(pm) = 1. Now (2.7) shows us that except for the ramified prime factors, ψC(x) is a partial sum of the coefficients of FC(s). Now let σ0 > 1, x ≥ 2 and define ∫ 1 σ0+iT xs IC(x, σ0, T ) = FC(s) ds. (2.8) 2πi σ0−iT s 2.2.1 Difference between ψC(x) and IC(x, σ0, T ) Lagarias and Odlyzko in [15, (3.8)-(3.11)] proved that for Re(s) > 1 and T > 0, ∣∣∣∣ ∑ ∣ ∣ I (x, σ , T )− θ(pm) log(N p)∣∣ ≤ n σ −1C 0 K 0T + nK(log x) +R0(x, σ0, T ), p,m N(pm)≤x and ∣∣∣ ∑ ∣∣ ∣θ(pm) log(N p)− ψC(x)∣∣ ≤ 2(log x)(log dL), p,m N(pm)≤x where ∑ ( )σ ( ∣∣∣ ∣ )x 0 −1∣ x ∣∣ −1 R0(x, σ0, T ) = min 1, T log ∣ log(N p). (2.9)N pm N pm p,m N(pm) =x Using this they obtained that ψC(x) = IC(x, σ0, T ) +R1(x, σ0, T ), where R1(x, σ0, T ) ≤ 2(log x)(log dL) + nKσ T−10 + nK(log x) +R0(x, σ0, T ). We prove an explicit version of their result. The key is to use Perron’s formula [6, Page 109-110]: 24 2.2. FORMULA FOR ψC(X) Lemma 2.2. If y > 0, σ > 0 and T > 0, then ∣∣∣ ∫ ∣1 σ+iT ys ∣∣ σ −1 −1 ∣∣ ds− 1∣∣ ≤ y min(1, T | log y| ) if y > 1,∣ 2πi σ−iT s∣∣ ∫ 1 σ+iT ys 1 ∣ ds− ∣∣ ∣∣ ≤ σT−1 if y = 1, 2πi∣ σ−iT s 2∣∣ ∫ 1 σ+iT ys ds 2πi − s ∣∣ ∣ ≤ yσ min(1, T−1| log y|−1) if 0 < y < 1. σ iT Lemma 2.3. Let x ≥ 2 and T > 0. With the above notations, we have |ψC(x)− IC(x, σ0, T )| ≤ R1(x, σ0, T ), (2.10) where ( ) ≤ 2 (log x)(log dL) 1R1(x, σ0, T ) | | + nK(log x) + σ0T −1 +R0(x, σ0, T ). (2.11) log 2 G 2 Proof. We notice that ∣ ∑ ∣ ∣ ∑ ∣ |ψC(x)− ∣ ∣ ∣ ∣ IC(x, σ0, T )| ≤ ∣∣ θ(pm) log(N p)− ψ (x)∣∣+ ∣C ∣I mC(x, σ0, T )− θ(p ) log(N p)∣∣. p,m p,m N(pm)≤x N(pm)≤x (2.12) Since the Dirichlet Series FC(s) in (2.7) is absolutely convergent for Re(s) > 1, we can integrate IC(x, σ0, T ) term by term to obtain ∫ 1 σ0+i+T xs IC(x, σ0, T ) = FC(s) ds 2πi ∫σ0−iT 1 σ0+iT ∑ s − xs= θ(pm)(log(N p))(N p) ms ds 2πi s ∑ σ0−iT p,m ∫1 σ0+iT ( )x sds = θ(pm)(log(N p)) . 2πi σ −iT (N p)m sp,m 0 Now we split at N(pm) = x and define: ∑ ∑ ∫ σ +iT ( )1 0 x sm ds ∑= θ(p )(log(N p)) − θ(pm) log(N p), 1 2πi (N p)m s p,m σ0−iT p,m N(pm)=x N(pm)=x 25 2.2. FORMULA FOR ψC(X) and ∑ ∑ ∫ σ +iT ( )1 0 x sds ∑ = θ(pm)(log(N p)) − θ(pm) log(N p). 2 2πi σ −iT (N p)m sp,m 0 p,m N(pm)= x N(pm) x and N(pm2 ) < x to obtain: ∣ ∣∣∣∑ ∣ ∣ ∣∣ = ∣ ∣∣ ∑ ∫ ∣∣ 1 σ0+iT ( )s ∑ ∣ θ(pm x ds )(log(N p)) − θ(pm) log(N p) 2 2πi − (N p)m s ∣∣p,m σ0 iT p,m ∣∣N(p∑ m) =x ∫ ( ) N(p∑m)x∣ ( ∫ ∣∣ ∑ 1 σ0+iT ( ) x s ) m ds ∣= θ(p )(log(N p)) − 1 ∣ 2πi (N p)m s ∣ p,m σ0−iT N( m∣∣p )x Now we use Lemma 2.2 for both y > 1 and y < 1 with y = x(N p)m and the fact that |θ(pm)| ≤ 1 to obtain ∣∣∣∑ ∣∣ ∣∣ ( ) ( ∣ ∣ ∑ x σ0 ≤ ∣min 1, T−1∣∣ xlog ∣ ∣∣−1)∣ log(N p) = R0(x, σm m 0, T ).2 N p N p p,m N(pm) =x (2.17) Therefore, using (2.13), (2.16) and (2.17), we get that for Re(s) > 1, ∣∣∣ ∑ ∣∣∣ ( )∣ 1I mC(x, σ0, T )− θ(p ) log(N p)∣ ≤ nK(log x) + σ T−10 +R0(x, σ0, T ). (2.18)2 p,m N(pm)≤x Now we focus on the first difference in Lemma 2.12. We notice that ∣∣∣∣ ∑ ∣ ∣ ∑ ∑ ∑ θ(pm) log(N p)− ψC(x)∣∣ ≤ log(N p) ≤ log(N p) 1. p,m p,m m N(pm)≤ p ramifiedx p ramified N(pm)≤x N pm≤x Serre [26, Proposition 5] proved ∑ ≤ 2log(N p) | log dL. (2.19)G| p ramified 27 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) Also we know that for each prime ideal p, N p ≥ 2. Hence, ∑ ≤ log x1 . log 2 m N(pm)≤x Therefore combining this with (2.19), we get ∣∣∣∣ ∑ ∣ ∣ ∑ θ(pm log x 2 (log x)(log dL) ) log(N p)− ψC(x)∣∣ ≤ log(N p) ≤ | | . (2.20)log 2 log 2 G p,m p ramified N(pm)≤x Now combining (2.18) and (2.20), we obtain the required result. Revision 1. Lagarias and Odlyzko in [15, (3.8)] and Winckler in [30, (1)] proved that for x ≥ 2, ∣∣∣ ∣∣ ∑ ∑IC(x, σ0, T )− ∣θ(pm) log(N p)∣∣ ≤ (log(N(p)) + σ −10T ) +R0(x, σ0, T ). p,m p,m N(pm)≤x N(pm)=x We corrected the sum on the right hand side of the above equation with ∑ ( )1 (log(N p)) + σ T−10 2 p,m N(pm)=x as shown in (2.14). 2.3 Estimating remainder terms R0(x, T ) and R1(x, T ) In this section, we will establish an estimate for R0(x, σ0, T ) as defined in (2.9). From here onwards we will fix σ0 = σ −1 0(x) = 1 + (log x) . We thus get x σ0 = ex. Since this fixed σ0 depends on x, we write R0(x, σ0, T ) as R0(x, T ), R1(x, σ0, T ) as R1(x, T ) and IC(x, σ0, T ) as IC(x, T ) from now on. We will write R0(x, T ) = S1 + S2 + S3, where S1 consists of those terms of (2.9) for which |N(pm)−x| ≥ 14x, S2 consists of those terms of (2.9) for which |N(pm)−x| ≤ 1, and S3 consists of the remaining terms of (2.9). Lagarias and Odlyzko in [15] proved bounds for S1, S2, S3 and therefore proved a non-explicit 28 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) bound for R0(x, T ). They proved that S1  nKxT−1 log x, S −1 22  nK log x and S3  nKxT (log x) . Putting all this together, they got, for x ≥ 2 and T ≥ 1, (see [15, (3.17)]) R0(x, T )  nK(log x) + n −1 2KxT (log x) . Using this they also obtained a bound for R1(x, σ0, T ) as defined in (2.11). They proved (see [15, (3.18)]) R1(x, T )  (log x)(log dL) + nK log x+ n xT−1K (log x)2. We try to produce explicit bounds for the terms S1, S2, S3, R0(x, T ) and R1(x, T ). 2.3.1 Bounding S1 We study ∑ ( )σ ( ∣ ∣ )x 0 ∣ x ∣−1 S1 = min 1, T −1∣ N pm ∣ log ∣∣ log(N p).N pm p,m |N(pm)−x|≥ 1x 4 To bound S1, we first prove a bound for the logarithmic derivative of the Dedekind ζ-function, ζK(s) : Lemma 2.4. For σ > 1, ζ ′ ζ ′− K (σ) ≤ −nK (σ) ≤ n (σ − 1)−1K . ζK ζ Proof. By Euler product formula for the Dedekind ζ-function, ζK(s) and the Riemann ζ- function, ζ(s), we obtain their logarithmic derivatives respectively as −ζ ′ ∑ K log(N p) ζ ′ ∑ log p (σ) = ζK (N p)σ − and − (σ) = − ,1 ζ pσ 1p p where in the second sum p runs through the rational primes. Also, for each prime ideal p, 29 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) N p = pk for some rational prime p and some positive integer k. Therefore log(N p) k log p k log p log p − =(N p)σ 1 pkσ − = × ≤ .1 p(k−1)σ + ...+ 1 pσ − 1 pσ − 1 Note that there are at most nK distinct p lying over a given rational prime p. Hence we get, ζ ′ ∑ log(N p) ∑− K ≤ log p ζ ′(σ) = nK ζK (N p)σ − 1 pσ − = −nK (σ).1 ζp p Using Abel’s summation formula for ζ(σ), we obtain σ ζ(σ) = − − σI(σ), (2.21)σ 1 ∫∞ where I(σ) = 1 (t− [t])t−σ−1dt. Taking the logarithmic derivatives on both sides, we get ′ ( − )ζ 1 1 (σ) = − − I(σ)− σI ′(σ) . (2.22) ζ ζ(σ) (σ 1)2 Now using the (2.21), (2.22) and the fact that I ′(σ) ≤ 0 for σ > 1, we get ζ ′ ( ) 1 1 −1 1 (σ) + − ≥ ( − − I(σ) +ζ σ 1 ζ(σ) (σ 1)2 σ − 1− )1 1 ζ(σ) 1 ζ(σ) = ζ(σ) (σ − + − +1()2 σ σ − 1 σ −) 1 1 − σ 2σ − 1= − − + ζ(σ) . (2.23)ζ(σ)(σ 1) σ 1 σ The Laurent series expansion of the Riemann zeta function about s = 1 is given by ∑∞1 (−1)nγn ζ(s) = − + (s− 1) n, s 1 n! n=0 where the constants γn are called the Stieltjes constants and can be defined by the limit ((∑m ) )(log k)n − (logm)n+1γn = l m→im∞ .k n+ 1 k=1 Notice that for all σ > 0, γ ≥ σ−10 2σ−1 . Now using this with the Laurent series expansion of ζ and 30 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) writing σ2 = (σ − 1)2 + (2σ − 1), we obtain that for σ > 1, ( ) σ2 ∑∞ n− − (−1) γn n+1 (σ − 1)2(σ 1)ζ(σ) − = 1 + γ0(σ − 1) + (σ − 1) − + 12σ 1 n! 2σ − 1 ∑ n=1∞ − (−1) nγ 2n = γ (σ 1) + (σ − 1)n+1 − (σ − 1)0 n! 2σ − 1 n=1 ≥ 0. (2.24) Therefore we get, 2 ζ(σ) ≥ σ , (σ − 1)(2σ − 1) and thus 2σ − 1 σ ζ(σ) ≥ − . (2.25)σ σ 1 Now using (2.25) in (2.23), we get that ζ ′ 1 (σ) + ζ σ − ≥ 0.1 Therefore, ζ ′−nK (σ) ≤ n (σ − 1)−1K . ζ Lemma 2.5. Let x ≥ x ≥ 2, T ≥ 1 and σ −10 = 1 + (log x) . Then ≤ (eS1 )xT−1nK(log x). log 54 Proof. Since |N(pm)− x| ≥ 14x, thus ∣∣∣ ∣∣ x ∣ 5log ∣ ≥ log ,N(pm) ∣ 4 and therefore ( ∣∣ ∣∣−1) ( ( ))−1 min 1, T−1∣∣ x 5log ∣ ≤ T−1 log .N(pm) ∣ 4 31 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) Now using this above result, xσ0 = ex and Lemma 2.4, we obtain that (− ) ∑ ( )1≤ xT e −mσ xT(−1e ζ ′S 01 N(p) log(N p) = ) − K (σ0) ζ log 5 p,m4 ( ) log 5 K 4 ≤ exT −(1n) ζ ′ −1K − ≤ exT( n)K − e(σ0) (σ0 1)−1 = ( )xT−1nK(log x). (2.26) log 5 ζ 4 log 5 4 log 5 4 2.3.2 Bounding S2 Now we will bound ∑ ( )x σ (0 − ∣1∣∣ ∣ )∣ x ∣ −1 S2 = min 1, T log ∣ log(N p). N(pm) N(pm) ∣ p,m N(pm) =x |N(pm)−x|≤1 Lemma 2.6. Let x ≥ x0 ≥ 2. Then S2 ≤ 2a1nK(log x+ a2), (2.27) ( )1+ 1 ( ) log(x0) where a1 = x0 x −1 and a2 = log 1 + 1 0 x . 0 Proof. Notice that S2 consists of those terms p m for which 0 < |N(pm)−x| ≤ 1. Also, there are at most 2n of such pmK . Together with ( ∣∣ ∣∣x −1) min 1, T−1∣∣ log ∣∣ ≤ 1 and x− 1 ≤ N(pm) ≤ x+ 1,N(pm) we obtain, ( ) x σ0 S2 ≤ 2nK log(x+ 1) − . (2.28)x 1 32 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) ( )σ0 Now we study the function log(x+ 1) xx−1 . Notice that ( )σ ( ( ))( )x 0 1 x σ0 log(x+ 1) − = log x(+ logx 1 )1 + x( x− 1 x σ ( ))( ) 0 1 x σ0 = (log x) − + log 1 + − . (2.29)x 1 x x 1 Now the second term ( ( ))( ) 1 x σ0 log 1 + x x− 1 is a d(ecreas)ing quantity for x ≥ x0 and achieves its maximum at x = x0 which is a1a2. Also,σ0 since xx−1 is decreasing for x ≥ x0 and its maximum value is at x = x0 which is a1, we get ( ) x σ0 log(x+ 1) − ≤ a1(log x+ a2).x 1 Now using this inequality in (2.28), we get the required result. 2.3.3 Bounding S3 Now we will bound ∑ ( )σ (x 0 − ∣∣ ∣−1)xS3 = min 1, T 1∣∣ logN pm N pm ∣ ∣∣ log(N p). p,m N(pm) =x 1<|N pm−x|< 1x 4 Lemma 2.7. Let x ≥ x0 ≥ 2 and T ≥ 1. Then S3 ≤ 5a n T−13 K x(log x)2, (2.30) 4 ( )2+ 1 log(x0) where a = 43 3 . Proof. We rewrite ∑ ( )σ ( ∣∣ ∣∣−1)x 0 x S3 = min 1, T −1∣∣ log ∣∣ log(N p).n n p,m n=N(pm)= x 1<|n−x|< 1x 4 33 2.3. ESTIMATING REMAINDER TERMS R0(X,T ) AND R1(X,T ) Notice that in the region where 1 < |n − x| < 14x, we get xn < 43 , n < 5x4 and max{x, n} < 4n3 . Using this and the mean value theorem, we get that for 1 < |n− x| < 14x, ∣∣∣ ∣−1∣ x ∣ 4 nlog ∣ ≤n ∣ 3 | − | . (2.31)x n ( )σ ( ) ( ∣0 σ0 ≤ − ∣∣∣ ∣ ∣−1) ∣∣ ∣∣−1 Now( us)ing x 4 , min 1, T 1 log x ∣∣ ≤ T−1∣∣ log x ∣n 3 n n ∣ , (log N p) ≤ log n ≤ log 5x4 , x ≥ x0 and (2.31), we get σ 10 + 1 ≤ 2 + log x and0 ( )σ ( ) ( )4 0+1−1 5x ∑ nS − ∑1 5x n3 ≤ T log ≤ a3T log . (2.32) 3 4 |x− n| 4 |x− n| p,m p,m n=N(pm) =x n=N(pm) =x 1<|n−x|< 1x 1<|n−x|< 1x 4 4 Now we can bound 1|x−n| by 1 |x−n| and make the variable c∑hange |x − n| = k. Considering|n− x| ≥ k where k ∈ Z and using n < 5x4 with the identity x 1k=1 k ≤ log(x) + 1 for x ≥ 1, we get ∑ n ∑ 5 x 5 ∑ ( ( ) ) | − | ≤ 1 5 x 2 = x ≤ x log + 1 . (2.33) x n 4 k 2 k 2 4 n k∈Z k∈Z 1<|n−x|< 1x 1 4 1≤k< x 1≤k< 1x4 4 R(ecal(l tha)t)th(ere a(re a)t mo)st nK prime ideals with N pm = n. Also it can be easily verified that log 5x log x4 4 + 1 ≤ (log x)2 for x ≥ 2. Thus, using (2.33) we obtain ( ( )) S ≤ a T−1 5x ∑ n ∑ 3 3 log 4 |x− | 1n n p,m ( ( )1<)|(n−x|<(1x N(pm)=n= x4 ( ) )) ≤ −1 5x 5 xa3nKT log x log + 1 4 2 4 ≤ 5a −1 23nKT x(log x) . (2.34) 2 Revision 2. Winckler in [30, (9)] showed that ∑ ( )n ∑ x | − | ≤ 1 + ,x n k n k 1<|n−x|< 1x 1 4 1 be the cyclic group generated by g, E be the fixed field of G0, and let χ denote the irreducible characters of G0. Since G0 is cyclic and the irreducible representations of a cyclic group are one-dimensional (i.e., the corresponding vector space is C), therefore the characters χ are one- 36 2.4. REDUCTION TO THE CASE OF HECKE L-FUNCTIONS dimensional (linear). Lemma 2.11. We have |C| ∑ L′ FC(s) = −|G| χ(g) (s, χ, L/E). (2.38)L χ Proof. Let τ : G0 → C be the class function defined by ⎪⎪⎧⎨ |G0| if h = g, τ(h) = ⎪⎩⎪ (2.39)0 if h = g. Now the orthogonality relations for characters of H imply that (since {g} is a conjugacy class of the cyclic group G0) ∑ τ = χ(g)χ. χ Recall that a left transversal of G0 is a subset of elements of G which contains exactly one element of each left coset of G0. Let τ ∗ denote the class function on G induced by τ . Recall that τ∗ is defined as τ∗ ∑ (y) = τ0(s −1ys) s∈S where S is a left transversal of G0 and τ0 is defined as τ0(h) = τ(h) if h ∈ G0 and 0 otherwise. Notice that with the above definition, if y ∈/ C, then τ∗(y) = 0. Now if y ∈ C, then y = a−1ga for some a ∈ G. Thus s−1ys = (as)−1g(as). Also if S is a transversal of G0 in G, then aS is a transversal of G0 in G. Moreover the class function τ ∗ is invariant under the transversals of G0 in G. Using this we see that for y ∈ C, τ∗ ∑ ∑ ∑ (y) = τ0(s −1ys) = τ −10((as) g(as)) = τ0((as)−1g(as)) = τ∗(g). s∈S s∈S as∈aS We know that τ0 is only non-zero at g (with value |G | at g) and s−10 gs = g if and only if s ∈ CG(g) where CG(g) is the centralizer of g in G. Therefore τ∗ ∑ (g) = τ (s−1 |CG(g)| 0 gs) = | × |G0| = |CG(g)|.∈ G0|s S 37 2.5. DENSITY OF ZEROS OF HECKE L-FUNCTIONS Therefore we obtain ⎪⎧⎨⎪ ∗ |CG(g)| if y ∈ C,τ (y) = ⎩⎪⎪ (2.40)0 if y ∈/ C. Now sinceG acts on itself by conjugation, therefore by orbit-stabilizer theorem, we get |CG(g)||C| = |G|. This gives us that τ∗ = fC (see (2.4)). This implies that ∑ χ(g)χ∗ ∑ = φ(g)φ, χ φ where χ∗ is the character of G induced by G0. Therefore for Re(s) > 1, (2.6) modifies into |C| ∑ L′ FC(s) = −| | χ(g) (s, χ ∗, L/K). (2.41) G L χ From the properties of L-functions as described in [18, Theorem 2.3.2 (d)], we get that L(s, χ∗, L/K) = L(s, χ, L/E). Therefore, for Re(s) > 1, (2.41) becomes |C| ∑ L′ FC(s) = −| χ(g) (s, χ, L/E),G| L χ and by analytic continuation of L-function, this holds for all s. 2.5 Density of zeros of Hecke L-functions Now combining (2.8) and (2.38), we obtain −|C| ∑ ∫1 σ0+iT xs L′ IC(x, σ0, T ) = | | χ(g) (s, χ, L/E)ds, (2.42)G 2πi χ σ0−iT s L where σ0 = 1 + (log x) −1 and χ runs through the (one-dimensional) irreducible characters of H =< g >. Our next goal is to evaluate each integrals in (2.42). To achieve this, we first prove some results relating to L′/L. Since L and E are going to be fixed from now on, we will use L(s, χ) to denote L(s, χ, L/E). 38 2.5. DENSITY OF ZEROS OF HECKE L-FUNCTIONS We let F (χ) denote the conductor of χ and set A(χ) = dE NE/Q(F (χ)) (2.43) and ⎪⎪⎧⎨ 1 if χ = χ1, the principal character, δ(χ) = ⎪⎩⎪ (2.44)0 otherwise. For each χ there exists non-negative integers a = a(χ) and b = b(χ) such that a(χ) + b(χ) = nE , (2.45) such that if we define ( ( ))b(χ)( ( ))a(χ) γ (s) = π− s+1 s+ 1 − s s χ 2 Γ π 2Γ (2.46) 2 2 and − δ(χ) sξ(s, χ) = (s(s 1)) A(χ) 2 γχ(s)L(s, χ), (2.47) then ξ(s, χ) satisfies the functional equation (see [18, Theorem 2.2.1]) ξ(1− s, χ) = W (χ)ξ(s, χ), (2.48) where W (χ) is a certain constant of absolute value one. This W (χ) is known as the root number. Furthermore, ξ is an entire function of order 1 and does not vanish at s = 0, and hence by the Hadamard product theorem we have (see [18, Theorem 2.4.1.1]) ∏( ) s ξ(s, χ) = e(B1(χ)+B(χ)s) − s1 e ρ (2.49) ρ ρ for some constants B1(χ) and B(χ), where ρ runs through all the zeros of ξ(s, χ), which are precisely the non-trivial zeros of L(s, χ) (i.e., zeros of L(s, χ) with 0 < (s) < 1). Recall that L(s, χ) and hence ξ(s, χ) have no zeros ρ with Re(ρ) ≥ 1. We are interested in the integral (2.42). Therefore we need to find identities involving L′/L. 39 2.5. DENSITY OF ZEROS OF HECKE L-FUNCTIONS We differentiate (2.47) logarithmically to obtain ′ ( )ξ 1 1 1 γ′ L′χ (s, χ) = δ(χ) + − + log(A(χ)) + (s) + (s, χ). (2.50)ξ s s 1 2 γχ L Similarly we differentiate (2.49) logarithmically to obtain ′ ∑( )ξ 1 1 (s, χ) = B(χ) + ξ s− + . (2.51)ρ ρ ρ Now combining (2.50) and (2.51), we obtain L′ ∑( ) ( )1 1 1 1 1 γ′χ (s, χ) = B(χ) + − + − δ(χ) + − − log(A(χ))− (s). (2.52)L s ρ ρ s s 1 2 γ ρ χ Theorem 2.12. With the notation as above, ∑ 1 Re(B(χ)) = − Re , (2.53) ρ ρ and L′ L′ ∑( ) ( )1 1 1 1 γ′χ (s, χ) + (s, χ) = − + − − log(A(χ))− 2δ(χ) + − 2 (s) (2.54)L L s ρ s ρ s s− 1 γ ρ χ holds identically in the complex variable s, where ρ runs through the non-trivial zeros of L(s, χ). Proof. Follows from [18, Corollary 2.4.1.2] and (2.54). ∣∣ ∣′ ∣ Lagarias and Odlyzko in [15, Lemma 5.2] proved that if Re(s) > 1, then ∣∣L ∣ nEL (s, χ)∣  Re(s)−1 . We give an explicit version to this ∣as∣ ∣∣ Lemma 2.13. If Re(s) > 1, then ∣∣L′ ∣ nEL (s, χ)∣ ≤ Re(s)−1 . Proof. The comparison of the corresponding Dirichlet series gives us ∣∣∣ ′ ∣∣L (s, χ)∣∣∣ ζ ′≤ − E (Re(s)).L ζE Applying Lemma 2.4 to this, we obtain the required result. 40 2.5. DENSITY OF ZEROS OF HECKE L-FUNCTIONS Lagarias and Odlyzko in [15, Lemma 6.1] proved that if |z + k| ≥ 18 for any non-negative integers k, then Γ′ (z)  log(|z|) + 2. Γ As an explicit version to this, Winckler in [30, Lemma 4.4] proved that : Lemma 2.14. 1. If Re(z) ≥ a, with a ≥ 1, then ∣∣∣∣Γ′ ∣∣ ∣ ∣ ≤ π 1(z) log(|z|) + + .Γ 2 a 2. If Im(z) ≥ b ≥ 1, then ∣∣∣ ′ ∣∣Γ ∣∣ ( ) (z)∣ ≤ 1 1log(|z|) + π 1 + + .Γ 2b 2b 3. If |z + k| ≥ 18 for every non-negative integers k, then ∣∣∣∣Γ′ ∣ ∣ (z)∣∣ ≤ log(|z|) + c1Γ where c = 831 5 . Lagarias and Odlyzko in [15, Lemma 5.3] proved that if Re(s) > −12 and |s| ≥ 18 , then ∣∣∣ ∣∣γ′χ (s)γ ∣ ∣∣  nE log(|s|+ 2). (2.55) χ Again as a explicit version to this, Winckler in [30, Lemma 4.5] proved: Lemma 2.15. If Re(s) > −12 and |s| ≥ 18 , then ∣∣∣ ′ ∣∣ ( )∣γχ nE(s)∣∣ ≤ log(1 + |s|) + c2 (2.56)γχ 2 where c = 1642 7 . Lagarias and Odlyzko in [15, Lemma 5.4] proved that if we let nχ(t) be the number of zeros ρ = β + iγ of L(s, χ) with 0 < β < 1 and |γ − t| ≤ 1, then for all t, we have nχ(t)  logA(χ) + nE log(|t|+ 2). (2.57) 41 2.5. DENSITY OF ZEROS OF HECKE L-FUNCTIONS As an explicit version to this, Winckler in [30, Lemma 4.6] proved: Lemma 2.16. Let nχ(t) be the number of zeros ρ = β + iγ of L(s, χ) with 0 < β < 1 and |γ − t| ≤ 1. For all t, we have ( ( )) nχ(t) + nχ(−t) ≤ c3 log(A(χ)) + nE log(|t|+ 3) + c4 (2.58) where c = 5 and c = 10753 2 4 134 . Revision 3. While going through the proof demonstrated by Winckler for [30, Lemma 4.6], we notice that the proof gives c3 = 5 instead of 5 2 . Lagarias and Odlyzko in [15, Lemma 5.5] proved that for all real such that 0 < ≤ 1, we have ∑ 1 B(χ) +  −1(logA(χ) + nE). | | ρρ < As an explicit version to this, Winckler in [30, Lemma 4.7] proved: Lemma 2.17. For all real such that 0 < ≤ 1, we have ∣∣∣ ∑ ∣∣ 1B(χ) + ρ ∣ ∣∣ ≤ c5 log(A(χ)) + c6nE |ρ|< ( ) ( ) where c = 1 5π2 10 10842 17905 8 + 34 +  and c6 = 107 + 157 . Lagarias and Odlyzko in [15, Lemma 5.6] proved that if s = σ + it with −12 ≤ σ ≤ 3 and |s| ≥ 18 , then ∣∣∣ ′ ∑ ∣∣L δ(χ) 1 ∣(s, χ) + − ∣− − ∣  logA(χ) + nE log(|t|+ 2).L s 1 s ρ ρ |γ−t|≤1 Winckler in [30, Lemma 4.8] proved an explicit version to this as : Lemma 2.18. If s = σ + it with −12 ≤ σ ≤ 3 and |s| ≥ 18 , then ∣∣∣∣L′ δ(χ) ∑ 1(s, χ) + − ∣ ∣ ( ) − − ∣ ∣ ≤ nE c9c7 log(A(χ)) + log(|t|+ 5) c8 + L s 1 s ρ 2 |t|+ 4 ρ |γ−t|≤1 + c10nE + c11, (2.59) 42 2.6. THE CONTOUR INTEGRAL ( ) where c = 5 1 + 7π2 , c = 57, c = 35, c = 50096 and c = 537 4 4 8 9 10 255 11 6 . 2.6 The contour integral In this section, we evaluate IC(x, T ) by evaluating ∫ σ ( )1 0+iT xs L′ Iχ(x, T ) = (s, χ) ds, (2.60) 2πi σ −iT s L0 for each character χ of H =< g >. From here onwards we impose an additional condition on T ≥ 1. T should not coincide with the ordinate of a zero of any of the L(s, χ). We also introduce a new parameter, U , which satisfies U = j + 12 for some non-negative integer j (with the aim of letting U → ∞). We define ∫ ( ′ )1 xs L Iχ(x, T, U) = (s, χ) ds, (2.61) 2πi B s LT,U where BT,U is the positively oriented rectangle with vertices at σ0 − iT , σ0 + iT , −U + iT and −U − iT . In this section we show that the difference Rχ(x, T, U) = Iχ(x, T, U)− Iχ(x, T ) (2.62) is small. To do this, we first divide the remainder Rχ(x, T, U) into one vertical integral and two horizontal integral. The vertical integral is given by ∫ 1 −T x− ( ) U+it L′ Vχ(x, T, U) = − (−U + it, χ) dt (2.63)2π T U + it L and the two horizontal integrals are given by ∫ −1/4( − ( ′ ) ( ))1 xσ iT L xσ+iT L′ Hχ(x, T, U) = − (σ − iT, χ) − (σ + iT, χ) dσ, (2.64)2πi −U σ iT L σ + iT L and ∫ ( ( ) ( )) ∗ 1 σ0 xσ−iT L′ xσ+iT L′ Hχ(x, T ) = (σ − iT, χ) − (σ + iT, χ) dσ. (2.65)2πi −1/4 σ − iT L σ + iT L 43 2.6. THE CONTOUR INTEGRAL In order to bound these integrals, we need the result relating to L′/L. Lagarias and Odlyzko in [15, Lemma 6.2] proved that if s = σ+it with σ ≤ −1/4, and |s+m| ≥ 1/4 for all non-negative integer m, then L′ (s, χ)  log(A(χ)) + nE log(|s|+ 2). (2.66) L As an explicit version to this, we prove Lemma 2.19. If s = σ + it with σ ≤ −1/4, and |s+m| ≥ 1/4 for all non-negative integer m, then ∣∣∣ ∣ ( )∣L′ ∣(s, χ)∣∣ ≤ log(A(χ)) + nE log(1 + |s|) + 4.452 + c1 , (2.67)L with c 831 = 5 . Proof. Now combining the logarithmic derivative of (2.47) and (2.48), we get L′ L′ γ′χ γ′χ (s, χ) = − (1− s, χ)− log(A(χ))− (1− s)− (s). (2.68) L L γχ γχ 44 2.6. THE CONTOUR INTEGRAL Since σ ≤ −1/4, therefore Re(1− s) ≥ 5/4. Thus we use Lemma 2.13 to obtain ∣∣∣∣L′ ∣ (1− ∣s, χ)∣ L ∣ ≤ nE nE− − ≤ 5 − = 4nE . (2.69)(Re(1 s)) 1 4 1 ∣∣ ∣∣ ∣∣ ∣∣ Now since |s+m| ≥ 1/4 for all non-negative integer m, therefore ∣∣ s2+m∣∣ ≥ 1 and ∣∣ s+18 2 +m∣∣ ≥ 18 . Thus from Lemma 2.14, it follows that ∣∣∣ ′( )∣Γ s ∣∣ ∣ (∣ ∣ ≤ ∣∣∣s ∣∣ ∣) ∣∣ ′( )∣∣ (∣∣ ∣∣)∣ Γ s+ 1 s+ 1log + c1, and ∣∣ ∣∣ ≤ log ∣∣ ∣∣ + c1. (2.70)Γ 2 2 Γ 2 2 Also, the logarithmic derivative of (2.46) gives us ∣∣∣∣γ′ ∣ ∣ ∣ ∣∣ n b ∣∣ ′ ( )∣ ∣Γ s+ 1 ∣∣∣ ∣ a ∣∣∣Γ′ ( )∣ χ E s ∣ (s) = log(π) + + ∣∣. (2.71)γχ 2 2 Γ 2 2 Γ 2 ∣∣ ∣∣ S∣ imilar∣ly, s∣ince |1 − s +∣m| ≥ 1/4 for all non-negative integer m, therefore ∣∣1−s ∣2 +m∣ ≥ 18 and ∣∣∣ ∣s +m∣∣ ∣= ∣∣ (1−s)+1 ∣2 2 +m∣∣ ≥ 18 . Thus from Lemma 2.14, it follows that ∣∣∣ ′( − )∣∣∣ (∣∣Γ 1 s ∣ ≤ log ∣∣ ∣) ∣1− s ∣∣∣ + c1. (2.72)Γ 2 2 Also, the logarithmic derivative of (2.46) gives us ∣∣∣ ′ ∣ ∣∣γχ − ∣∣∣ nE b ∣ ( ) Γ′ s ∣∣ ∣∣ ′( )∣a Γ 1− s ∣ (1 s) = log(π) + ∣ ∣+ ∣ ∣. (2.73) γχ 2 2 ∣ Γ 2 ∣ 2 ∣ Γ 2 ∣ 45 2.6. THE CONTOUR INTEGRAL Now combining (2.68) with (2.69), (2.70), (2.71), (2.72) and (2.73), we get ∣∣∣ ′ ∣∣ ∣∣ ′ ∣∣L γχ ∣ ∣∣ ∣γ′χ ∣ (s, χ)∣∣ ≤ 4nE + log(A(χ)) + ∣∣ (1− s)∣∣+ ∣∣∣ ((s) ∣ L γ γ ∣χ ∣ χ′ )∣ ∣ ( )∣≤ nE b Γ s ∣ a ∣Γ′ 1− s ∣4nE + log(A(χ)) +∣ (log(π))+ ∣ ∣ 2 ∣ 2 ∣ Γ∣ 2( ∣ ∣ ∣ ∣∣ ′ ∣∣ ∣∣ ′ ) +∣ 2 ∣ Γ 2 ∣ nE b ∣Γ s+ 1 a Γ s ∣+ log(π) + ∣+ ∣2 2 Γ 2 (2 ∣ Γ(∣∣∣ 2 ∣ ∣ ≤ nE s ∣) )nE(4 +( log((π∣)) + log(A(χ)) + log ∣ ∣ + c1 2 2 ∣ a ∣∣∣1− s ∣ ∣∣) ) ( (∣ ∣) )∣ b ∣∣∣1 + s ∣+ log + c1 + log( ( ∣ ∣ + c 2 2 2 2 )1) ≤ 1 + |s|nE(4 + c1 + log(π)) + log(A(χ)) + nE log ( 2 ) = nE(4 + c1 + log((π)− log 2) + log(A(χ)) +)nE log(1 + |s|) ≤ log(A(χ)) + nE log(1 + |s|) + 4.452 + c1 . Remark 2.20. Winckler in [30, Lemma 5.1] proved that under the same condition as that of Lemma 2.19, ∣∣∣ ′ ∣∣ ( )∣L (s, χ)∣L ∣ ≤ 19683log(A(χ)) + nE log(2 + |s|) + . (2.74)812 2.6.1 Bounding Vχ(x, T, U) and Hχ(x, T, U) Now we observe that since U = 12+j for some non-negative integer j, we get |−U+it+m| ≥ 1/4. Thus we employ Lemma 2.19 to bound Vχ(x, T, U) and Hχ(x, T, U). Lagarias and Odlyzko in [15, (6.8),(6.9)] showed that x−U Vχ(x, T, U)  T (log(A(χ)) + nE log(T + U)), (2.75) U and −1/4 Hχ(x, T, U)  x (log(A(χ)) + nE log T ). (2.76) T We prove explicit versions of (2.75) and (2.76). We have: Lemma 2.21. If x ≥ 2, T ≥ 1 and T does not coincide with the ordinate of a zero of any of 46 2.6. THE CONTOUR INTEGRAL the L(s, χ), then ( ) −U | x TVχ(x, T, U)| ≤ log(A(χ)) + nE(log(U + T + 1) + 4.452 + c1) , πU (2.77) and − [ ( ( ) )]x 1/4| 5Hχ(x, T, U)| ≤ log(A(χ)) + nE log + T + 4.452 + c1 πT (log x) 4 (n ) x−1/4E+ , (2.78) (log x)2 πT 54 + T where c1 is defined in Lemma 2.19. Proof. Using Lemma 2.19, we have |V (x, T, U)| = ∣∣∣ ∫∣ 1 − ∣ T − ( )x U+it L′ ∣ χ ∣− (−U∣ + it, χ) dt ∣ 2π −U ∫ ∣T ≤ x T ∣ U + it L∣L′ ∣(−U + it, χ)∣dt 2πU − ∫− ∣ T T (L ∣ ( ))U ≤ x ( log(A(χ)) + nE log(1 + |s|) + 4.452 + c1 ) dt2πU −T x−U≤ T log(A(χ)) + nE(log(U + T + 1) + 4.452 + c1) , πU and ∣∣ ∫ − ( − ( ′ ) ( ′ )) ∣∣1 1/4 xσ iT L xσ+iT| LHχ(x, T, U)| = ∣∣ − (σ − iT, χ) − (σ + iT, χ) dσ∣2πi∫ − ∣σ iT L ∣ σ + iT L ∣U 1 −1/4≤ ∣∣∣σ L′ ∣x (σ + iT, χ)∣∣dσπT ∫−U (L ( )) ≤ 1 −1/4 xσ log(A(χ)) + nE log(1 + |s|) + 4.452 + c∫ ( ( 1 dσ πT −U 1 −1/4 )) ≤ xσ log(A(χ)) + nE log(1 + |σ|+ T ) + 4.452 + c1 dσ πT ∫−U 1 −1/4 ( ( )) ≤ xσ log(A(χ)) + nE log(1 + |σ|+ T ) + 4.452 + c1 dσ. πT −∞ 47 2.6. THE CONTOUR INTEGRAL Now ∫ −1/4 (x− )1/4 (log(A(χ)) + nE(4.452 + c1)) x σdσ = (log(A(χ)) + nE(4.452 + c1)) , (2.79) −∞ log x and we employ integration by parts to obtain ∫ −1/4 ( ( )) −1/4 −1/4 xσ(log(1 + T + |σ|))dσ ≤ 5 x 1 xlog + T + ( ) (2.80) −∞ 4 log x (log x)25 4 + T Now combining (2.79), (2.79) and (2.80), we obtain − [ ( ( ) )]x 1/4 5 n x−1/4| EHχ(x, T, U)| ≤ log(A(χ)) + nE log + T + 4.452 + c1 + ( ) . πT (log x) 4 5 (log x) 2 πT 4 + T Remark 2.22. Winckler in [30, (30),(31)] proved that ( ( )) | | ≤ x −UT 19683 Vχ(x, T, U) log(A(χ)) + nE log(U + T + 2) + , (2.81) πU 812 and − [ ( ( ) )]x 1/4 9 19683 n −1/4| | ≤ E xHχ(x, T, U) log(A(χ)) + nE log + T + + ( ) . πT (log x) 4 812 9 (log x) 2 πT 4 + T (2.82) 2.6.2 Bounding H∗χ(x, T ) Now we give an estimate for H∗χ(x, T ). Lagarias and Odlyzko in [15, (6.12)] proved that ∗ x log xHχ(x, T )  (log(A(χ)) + nE log T ), (2.83)T 48 2.6. THE CONTOUR INTEGRAL and as a result, they showed in [15, (6.13)] that, Iχ(x, T )− Iχ(x, T, U) = −Vχ(x, T, U)−Hχ(x, T, U)−H∗χ(x, T )  x(log x) Tx −U (log(A(χ)) + nE log T ) + (log(A(χ)) + nE(log(T + U)). T U (2.84) We prove explicit versions of (2.83) and (2.84). Lemma 2.23. Suppose ρ = β+ iγ, with 0 < β < 1 and γ = t. If |t| ≥ 2, x ≥ 2 and 1 < σ1 ≤ 3, then ∣∣∣ ∫ ∣∣ σ1 xσ+it ∣ (12σ + 27)xσ11− dσ∣∣ ≤ | | − − . (2.85)−1/4 (σ + it)(σ + it ρ) ( t 1)(σ1 β) Proof. First we suppose that γ > t. Let B be the rectangle with vertices at σ1+ i(t−1), σ1+ it, −1 + it and −14 4 + i(t− 1), oriented counterclockwise. By Cauchy’s theorem, ∫ xs ds = 0, B s(s− ρ) since the integrand has no singularities inside the contour. Also, on the three sides of the rectangle other than the segment from −14+it to σ1+it, since 0 < σ1−β ≤ 3 and 14 < 14+β < 54 , therefore the integrand is majorized by 12xσ1 (2.86) (|t| − 1)(σ1 − β) which proves the result for γ > t. A similar proof for γ < t uses the rectangle with vertices at σ1 + i(t+ 1), σ1 + it, −14 + it and −14 + i(t+ 1). σ Revision 4. Winckler in [30, Lemma 5.2] stated that the x 1(|t|−1)(σ −β) is enough to majorize1 σ the corresponding integrands majorized by 12x 1(|t|−1)(σ −β) as in (2.86) which is not correct and we1 corrected it by m∣∣ ultiply∣∣ing a factor of 12 to the numerator as shown in (2.86). For example,s 3 taking σ1 = 3, ∣ x 1 1 xs(s−ρ) ∣ in the segment −4 + i(t − 1) to −4 + it, is majorized by (|t|−1)( 1 .+β) 4 Using Winckler’s assumption, we have (3− β) ≤ 14 + β equivalent to 2β ≥ 114 which is incorrect because β < 1. 49 2.6. THE CONTOUR INTEGRAL Lemma 2.24. If x ≥ 2, T ≥ 1 and T does not coincide with the ordinate of a zero of any of the L(s, χ), then − − ( ( ) )| ∗ ex x 1/4 nE c9 δ(χ)Hχ(x, T )| ≤ c7 log((A(χ()) + log(T + 5() c8 + +πT (log x) 2 T + 4 )))T e (12σ0 + 27)x + − − c3 log(A(χ)) + nE log(T + 3) + c4 . (2.87)2π (T 1)(σ0 1) Proof. Notice that for σ ∈ [−14 , σ0], x ≥ 2 and T ≥ 2, Lemma 2.18 gives us ∣∣∣ ′ ∑ ∣∣∣ ( )∣L − 1 ∣ ≤ nE c9 δ(χ)(σ + iT, χ) − c7 log(A(χ)) + log(T + 5) c8 + + ,L σ + iT ρ 2 T + 4 T ρ |γ−T |≤1 (2.88) and ∣∣∣ ∣ ( )∣L′ ∑ 1 ∣ nE c9 δ(χ)(σ − iT, χ)− ∣− − ∣ ≤ c7 log(A(χ)) + log(T + 5) c8 + + .L σ iT ρ 2 T + 4 T ρ |γ+T |≤1 (2.89) 50 2.6. THE CONTOUR INTEGRAL Therefore using (2.88) and (2.89), we get ∣∣∣ ∫ (∣ σ∗ 1 0 xσ−iT ∑ ) − 1 x σ+iT ∑ 1 H (x, T ) − dσ∣∣∣χ 2πi −1/4 σ − iT σ − iT − ρ σ + iT σ + iT − ρ ∣ ∣ ρ ρ∣ |γ+T |≤1 |γ−T |≤1 = ∣ ∫ σ ( (∣ 1 0 xσ−iT L′ ∑ ) 1 2πi −1/4 σ − (σ − iT, χ)− iT L σ − iT − ρ ρ ( |γ+T |≤1 )) ∣ xσ+iT L′ ∑− 1 ∣(σ + iT, χ)− dσ∣ σ + iT L σ + iT − ρ ∣ ∫ (∣ ρ 1 σ0 xσ ∣∣∣L′ ∑ |γ−T |≤1 ≤ 1(σ − iT, χ)− ∣∣∣ 2π −1/4 T L σ − iT − ρ ∣ ∣ ρ∣ |γ+T |≤1 ∣)L′ ∑ 1 ∣ + ∣∣ (σ + iT, χ)− ∣ dσL σ + iT − ρ ∣ ( ( ρ|γ−T |≤1) )∫ 1 n c δ(χ) σ0≤ E 9c7 log(A(χ)) + log(T + 5) c8 + + xσdσ πT ( 2 T(+ 4 T) −1/4− − )ex x 1/4 nE c9 δ(χ) = c7 log(A(χ)) + log(T + 5) c8 + + . (2.90) πT (log x) 2 T + 4 T Now using Lemma 2.23, we get ∣∣∣∣ ∫ ( ) ∣ ∣ σ ( )∣1 0 xσ−iT ∑ ∣ ∑ ∫ σ0 σ−iT∣ 1 ∣ ∣∣ 1 x ∣− dσ∣ = dσ ∣2πi −1/4 σ iT σ − iT − ρ ∣ ∣2πiρ ρ − ∣1/4 (σ − iT )(σ − iT − ρ) |γ+T |≤1 |∑γ+T |≤∣1∫ ∣ ≤ 1 2π ∣ ∣∣ σ0 xσ−iT ∣dσ∣ −1/4 (σ − iT )(σ − iT − ρ) ∣ρ |γ+T |≤1 ≤ 1 ∑ (12σ0 + 27)xσ0 2π (T − 1)(σ0 − β)ρ |γ+T |≤1 ≤ 1 (12σ0 + 27)x σ0 ∑ 1 2π (T − 1)(σ0 − 1) ρ |γ+T |≤1 1 (12σ0 + 27)x σ0 = − − nχ(−T ). (2.91)2π (T 1)(σ0 1) Similarly, we get ∣∣∣ ∣∣ ∫ ( )1 σ0 xσ+iT ∑ 1 ∣∣ ∣ 1 (12σ0 + 27)x σ0 − dσ∣ ≤ nχ(T ). (2.92)2πi −1/4 σ + iT σ + iT ρ ∣ 2π (T − 1)(σ0 − 1)ρ |γ−T |≤1 51 2.6. THE CONTOUR INTEGRAL Now combining (2.90), (2.91), (2.92) and Lemma 2.16, we get − ( ( ) )| ∗ | ≤ ex x−1/4 nE c9 δ(χ)Hχ(x, T ) c7 log(A(χ)) + log(T + 5) c8 + +πT (log x) 2 T + 4 T 1 (12σ0 + 27)x σ0 + − ( − (nχ(T ) + nχ(−T ))2π (T 1)(σ0 1)− − ( ) )≤ ex x 1/4 nE c9 δ(χ)c7 log(A((χ)) + log(T +(5) c8 + +πT (log x) 2 T + 4 )) T e (12σ0 + 27)x + − − (c3 log(A(χ)) + nE log(T + 3) + c4 ).2π (T 1)(σ0 1) 2.6.3 Bounding Rχ(x, T, U) Lemma 2.25. If x ≥ 2, T ≥ 1 and T does not coincide with the ordinate of a zero of any of the L(s, χ), then ( ) x−U| | ≤ TRχ(x, T, U) log(A(χ)) + nE(log(U + T + 1) + 4.452 + c1) πU − [ ( ( )x 1/4 5 + (1− c7) log(A(χ))(+ nE log ))+ T + 4].452πT (log x) 4 − 1 c9 − δ(χ)+ c1 log(T + 5) c8 + 2 T + 4 T + (nE ) x−1/4 5 (log x) 2 πT 4 + T( ( ) ) e x nE c9 δ(χ) + c7 l(og(A(χ)) + lo(g(T + 5) c8 + +π T (log x) 2 T)+)4 T 39e x(log x) + − ((c3 log(A(χ)) +(nE log(T + 3) + c4 )2π T 1 )) 6e x + − (c3 log(A(χ)) + nE log(T + 3) + c4 ). (2.93)π T 1 Proof. From (2.62), remember that |Rχ(x, T, U)| = |Iχ(x, T, U)− Iχ(x, T )| ≤ |Vχ(x, T, U)|+ |Hχ(x, T, U)|+ |H∗χ(x, T )|. Now combining (2.77), (2.78) and (2.87) with the fact that σ = 1+(log x)−10 , we get the required result. 52 2.7. THE EXPLICIT FORMULA Remark 2.26. Winckler in [30, (32)] showed that ( | ∗ | ≤ 5(13(log x) + 4)ex 571 ex− − ) ((x 1/4 log(A(χ)) nE 57 ex− x−1/4 Hχ(x, T ) + +8(T − 1) ) 25 T (log x) π π 2 T (log x) ) 5(13(log x) + 4)ex 5375(13(log x) + 4)ex 5921 ex− x−1/4 + − log(T + 5) + + ,8(T 1) 2144(T − 1) 28 T (log x) and thus in [30, (33)] showed that [ ( )] | − | ≤ 65e x(log x) 1075Iχ(x, T ) Iχ(x, T, U) − [ log(A(χ)) + nE( log(T + 5) +8π (T 1) 268)] 5e x 1075 + − [ log(A(χ)) + nE lo(g(T + 5) +2π (T 1) 268 )] e x 571 57 5921 + [ log(A(χ()) + nE log(T + 5) +π T (log x) 25 2− )2]8x UT 19683 + log(A(χ)) + nE log(U + T + 2) + πU 812 4n x−1/4E + . 17πT (log x)2 2.7 The explicit formula 2.7.1 Estimating Iχ(x, T, U) Recall that the integral Iχ(x, T, U) is defined in (2.61). We first evaluate this integral. We recall that x ≥ 2, U = j + 12 for some non-negative integer j, and T ≥ 2 does not equal the ordinate of any zero of any of the L(s, χ). By Cauchy’s residue theorem, Iχ(x, T, U) equals the sum of the residues of the integrand at poles inside BT,U . Now if χ = χ1, the principal character, then L′/L has a first order pole of residue −1 at s = 1, and hence (this term being absent if χ = χ1) we obtain a contribution of −δ(χ)x from the possible pole at s = 1. Further, L′/L has a first order pole with residues +1 at each non-trivial zero∑ρ of L(s, χ) (the ρ’s areρ counted according to their multiplicity), and so such ρ’s contribute xρ ρ . In addition, L ′/L has first order poles at the so-called trivial zeros, which are real and non-positive. Also, the functional equation of L(s, χ) in (2.47) and study of Γ-function shows that L′/L has first order poles at s = −(2m− 1), m = 1, 2, ... where the residue is b(χ), and first order poles at s = −2m, m = 0, 1, 2, ... where the residue is a(χ). Therefore, the residues at points s with Re(s) < 0 53 2.7. THE EXPLICIT FORMULA contribute [ ] [ ] ∑U+1 U2 2x−(2m−1) ∑ x−2m−b(χ) − − a(χ) .2m 1 2m m=1 m=1 Now the only remaining residue is at s = 0, where we have the complication that both xs/s and L′ s /L may have first order poles. The Laurent series expansions of xs about s = 0 gives us xs 1 = + log x+ sh1(s), s s where h1(s) is a function that is analytic at s = 0. Similarly, the Laurent series expansion of L′/L as defined in (2.52) about s = 0 shows that L′ a(χ)− δ(χ) (s, χ) = + r(χ) + sh2(s), L s where h2(s) is a function that is analytic at s = 0 and ′( ) − 1 nE b(χ) Γ 1 a(χ) Γ ′ r(χ) = B(χ) (logA(χ)) + (log π) + δ(χ)− − (1). (2.94) 2 2 2 Γ 2 2 Γ s ′ Therefore the residue of x Ls L (s, χ) at s = 0 is r(χ) + (a(χ) − δ(χ)) log(x). Now, combining all these residue terms, we get [ ] [ ] ∑ xρ ∑ U+1 U 2 2 x1−2m ∑ x−2m Iχ(x, T, U) = −δ(χ)x+ − b(χ) − a(χ) + r(χ) + (a(χ)− δ(χ)) log x. ρ 2m− 1 2m ρ | | m=1 m=1γ 0 such that T + does not coincide with the ordinate of a zero of any of the L(s, χ) , 60 2.7. THE EXPLICIT FORMULA then ∣∣∣ | | ∣∣ − C ∣ψC(x) | |x+ S(x, T )∣G ∣ ≤ (log x)(log dL)2.89( | | + 13.83nK(log x) + 28.7n −1 2 [ ( K (T + ) x(log x) + 4.41n ( ) KG|C| x−1/4 5 + | | (1− c7) log dL + nL log + (T + ) + 4.452G π(T + )(log x) ( )) 4 ] 1 c 1 (n ) x−1/4− 9 L+ c1 log((T + ) + 5) c8 + − + 2 (T + ) + 4 T + 5 (log x) 2 ( ( π(T + ) )4 + (T +) ) e x nL c9 1 + c7 log dL + log((T + ) + 5) c8 + + π (T + )(log x) ( 2 ( )(T)+ ) + 4 T + 39e x(log x) + − (c(3 log dL + nL( log((T + ) + 3) + c)4 )2π (T + ) 1 ) 6e x + ( ( − (c3 (log dL + nL log()(T)+) ) + 3) + c4 )π (T + ) 1 x + c3 log dL + nL log(T + 3) + c4 T ( ) ( ) ) 1 log π γ log 3 + nL(log x) + c5 + log dL + nL + + log 2 + + c6 + 1 . (2.107) 2 2 2 2 where S(x, T ) is defined as in (2.100). Proof. Let T = |γ| for some ρ = β + iγ where 0 < β < 1. Now we evaluate (2.101) with T replaced by T + for the very clo∣∣se to zero and T + not co∣∣inciding with the ordinate of a zero of any of the L(s, χ) to obtain ∣∣ψ (x)− |C|C |G|x+ S(x, T + )∣∣. We also notice that | ( )C| ∑ ∑ xρ ∑ 1 |C| ∑ ∑ xρ S(x, T + ) = | | χ(g) − = S(x, T ) +G ρ ρ | | χ(g) . (2.108)G ρ χ ρ ρ ρ|γ| χ 1, where α(m) ≥ 0 for all m. Hence ( ′ ) ∞ − ζL − ζ ′ ζ ′ ∑L α(m)Re 3 (σ) 4 (σ + it)− L (σ + 2it) = (3 + 4 cos(t logm) + cos(2t logm)). ζL ζL ζL mσm=1 63 2.8. ZERO-FREE REGIONS We also know that 3 + 4 cos θ + cos 2θ = 2(1 + cos θ)2 ≥ 0. Therefore combining this with the fact that α(m) ≥ 0 and mσ > 0, we obtain that ( ) − ζ ′ ′ L ζ ζ ′ Re 3 (σ)− 4 L (σ + it)− L (σ + 2it) ≥ 0. (2.112) ζL ζL ζL Now if we consider the trivial normal extension L of L, then ζL(s) is the Artin L-function associated to the principal character. Let γL(s) denote the gamma factor associated to ζL(s). Then, (2.54) gives us ζ ′ ∑( )L 1 1 2 2 γ′2 (s) = − + − − log d LL − −ζL s ρ s ρ s s− − 2 (s), (2.113)1 γρ L where the summation is over the non-trivial zeros ρ = β+iγ of ζL(S). Since 0 < β < 1, therefore if Re(s) > 1, then Re(s− ρ)−1 > 0 for each zero ρ. Therefore, for 2 ≥ σ > 1, using Lemma 2.15 and (2.113), we obtain ( ′ ) (∑( )) ′ − ζRe L −1 1 1 1 1 1 γ(σ) = Re − + − + log dL + + − + L (σ) ζL 2 σ ρ σ ρ 2 σ σ 1 γρ L ≤ 1 1 1 γ ′ log d + + LL 2 σ σ − + ((σ)1 γL ) ≤ 1 1 nLlog dL + 1 + − + (log 3) + c2 . (2.114)2 σ 1 2 Let ρ′ = β + iγ be some particular zero with |γ| ≥ (1 + 4 log d )−1L . Now combining (2.15) and 64 2.8. ZERO-FREE REGIONS (2.113), we get that for 2 ≥ σ > 1, ( ′ ) (∑( )) − ζ 1 1 1 1Re L (σ + 2iγ) =− Re + + log dL ζL 2 ( σ + 2iγ − ρ )σ + 2iγ(− ρ ) 2ρ 1 1 γ′ +Re ( + − +Re L (σ) σ + 2iγ σ + 2iγ 1 )γL ( ′ ) ≤1 1 1 γlog d LL +Re + 2 (σ + 2iγ σ + 2iγ − )+Re( (σ)1 γL ) ≤1 1 1 nLlog dL +Re( +2 σ + 2iγ σ + 2iγ − )+ ((log(2|γ|+ σ + 1)) + c21 2 ) ≤1 1 1 nLlog dL +Re + − + (log(2|γ|+ 3)) + c2 .2 σ + 2iγ σ + 2iγ 1 2 (2.115) Since |γ| ≥ (1 + 4 log dL)−1, hence for 2 ≥ σ > 1, ( ) 1 1 1 1 1 3 Re + − ≤ + | | ≤ 1 + + 2 log dL = + 2 log dL.σ + 2iγ σ + 2iγ 1 σ 2 γ 2 2 Now combining this with (2.115), we obtain ( ′ ) ( ) − ζRe L 5 nL 3(σ + 2iγ) = log dL + (log(2|γ|+ 3)) + c2 + . (2.116) ζL 2 2 2 Now, again combining (2.15) and (2.113), we also get that for 2 ≥ σ > 1, ( ζ ′ ) − Re L((σ + i(γ)ζL )) ( ) ( 1 ∑ 1 1 1 1 1 γ′ ) = − Re − + + log d +Re + + Re L L (σ) 2 (σ + iγ ρ σ + iγ −)ρ (2 ) (σ + iγ σ)+ iγ − 1 γρ L ≤ 1 1 1 γ ′ 1 log dL +Re( + L − )+Re( (σ))− Re2 σ + iγ σ + iγ 1 γL σ + iγ − ρ′ 1 1 1 γ′ 1 = log d LL +Re + + Re (σ) − . 2 σ + iγ σ + iγ − 1 γL σ − β (2.117) Again since |γ| ≥ (1 + 4 log d )−1L , hence for 2 ≥ σ > 1, ( ) 1 1 1 1 Re + − ≤ + | | ≤ 1 + 1 + 4 log dL = 2 + 4 log dL.σ + iγ σ + iγ 1 σ γ 65 2.8. ZERO-FREE REGIONS Now combining this with (2.117) and (2.15), we obtain ( ′ ) ( ) − ζL 9 nL 1Re (σ + iγ) = log dL + (log(|γ|+ 3)) + c2 + 2− − . (2.118)ζL 2 2 σ β Now combining (2.112), (2.114), (2.116) and (2.118), we get ( ) 4 3 5 log 54 25 − < − + 22(log dL) + nL (log(|γ|+ 3)) + 4c2 + + . (2.119)σ β σ 1 2 2 2 ( ) Now suppose we take l = 22(log d ) +n 5(log(|γ|+3))+ 4c + log 54 25L L 2 2 2 + 2 and let σ = 1+ al . Thus (2.119) turns out to be 4l 3l − < + l,(1 β)l + a a which on further computing turns out to be equivalent to a− a2 β < 1− . (3 + a)l 2 √ Further, the evaluation of the function f(a) = a−a(3+a)l shows that f is maximum at 2 3− 3 and√ √ √ f(2 3− 3) = 7−4 3l . Now using a = 2 3− 3, we obtain √ ( ( ) )5 log 54 25 −1 σ = 1 + (2 3− 3) 22(log dL) + nL (log(|γ|+ 3)) + 4c2 + + , 2 2 2 and using this σ in (2.119), we obtain the required result. Remark 2.33. Winckler in [30, Lemma 7.1] proved that the ζL function has no zeros ρ = β + iγ in the region |γ| ≥ (1 + 4 log d )−1L √ ( ( ) )−1 β ≥ − − 5 1078 151 (7 4 3) 22(log dL) + nL (log(|γ|+ 3)) + + 2(log 3) + . 2 67 2 Revision 7. The constant 152 as given in [30, Lemma 7.1] by Winckler is not correct and we corrected this mistake by replacing that value with 252 as shown in Lemma 2.32. Lemma 2.34. [15, Lemma 8.2]. If nL > 1, then ζL(s) has at most one zero ρ = β + iγ in the 66 2.8. ZERO-FREE REGIONS region |γ| ≤ (4 log d )−1L and β ≥ 1− (4 log d )−1L . (2.120) Proof. (2.113) shows that 1 < σ ≤ 2, ∑ σ − β 1 1 ζ ′L 1 γ′ − = − + log d L L + (σ) + + (σ). (2.121) (σ β)2 + γ2 σ 1 2 ζ σ γ ρ L L ( ) ( ) ′ ′ ′ Notice that ζζ (σ) ≤ 0. Also, for 1 < σ ≤ 2, Γ σ < 0, Γ σ+1Γ 2 Γ 2 < 0 and since nL > 1, therefore 1 − nLσ 2 log π < 0. Hence using the logarithmic derivative of (2.46), we get ′ ( ) ′( ) ′( )1 γL 1 − nL a(L) Γ σ b(L) Γ σ + 1+ (σ) = log π + + < 0. (2.122) σ γL σ 2 2 Γ 2 2 Γ 2 Now using this, (2.121) gives us ∑ σ − β 1 1 (σ − < + log dL. (2.123)β)2 + γ2 σ − 1 2 ρ Due to symmetry of zeros about the real axis for ζL(s), presence of one complex zero of ζL(s), ρ = β + iγ with γ = 0 guarantees the presence of another zero. If ρ = β + iγ is a complex zero (i.e. |γ| = 0) in the region described by (2.120), then there are at least two zeros in that region and (2.121) gives us σ − β 1 1 2 − ≤ − + log dL. (2.124)(σ β)2 + γ2 σ 1 2 Now let us choose σ = 1 + (log d )−1L ≤ 2. Then 1 1σ−1 + 2 log dL = 32 log dL. Let ρ = β + iγ be in the region described by (2.120) and γ = 0. We thus set β = 1 − alog d with 0 < a ≤ 1/4L and |γ| = blog d with 0 < b ≤ 1/4. We now consider the function f given by f(a, b) = 2(a+1)L (a+1)2+b2 and use multi-variable calculus to find out that f attains the minimum value in the region 0 < a ≤ 1/4 and 0 < b ≤ 1/4 at a = 1/4 and b = 1/4 and the minimum value is approximately 67 2.8. ZERO-FREE REGIONS 1.53846. Therefore, a+1 σ − β × ( )log dL( ) 2(a+ 1)2 = 2 = log dL (σ − β)2 + γ2 2 2 (a+ 1)2 + b2 a+1 b log d +L log dL ≥ 1 11.53846 log dL > 1.5 log dL = σ − + log dL,1 2 and thus we get a contradiction to (2.123) in the case where γ = 0. Hence ζL(s) cannot have a complex zero in the region described by (2.120). Now suppose ζL(s) has more than one real zero (i.e. γ = 0) or a double real zero in the region defined by (2.120), say ρ1 = β1 and ρ2 = β2. Let β = min{β1, β2}. Then (2.123) together with 5 14 log d ≥ σ − β >L log d givesL ∑ σ − β σ − β1 σ − β2 1 1 > + = + (σ − β)2 + γ2 (σ − β1)2 (σ − β2)2 σ − βρ 1 σ − β2 ≥ 2 ≥ 2× 4− log dL = 1.6 log dL > 1.5 log dL,σ β 5 which is again a contradiction to (2.123). Thus there cannot be more than one zero in the region described in (2.120) and if such a zero exists, then it must be real and simple. Remark 2.35. If the possible zero described by the above lemma exists, we denote it by β0 and call it the exceptional (Siegel) zero. We also note that if nL = 1 (so that L = Q, log dL = 0), then ζL has no non-trivial zeros ρ = β + iγ with |γ| < 14. If β0 exists, then (2.110) shows that there exists a unique χ0 such that L(β0, χ0) = 0. This χ0 must then be a real character, since L(β0, χ0) = L(β0, χ0) = 0. Remark 2.36. Many number theorists have proved several results related to the zeros of Dedekind ζ-function. For example, Stark [27] in 1974 also proved that ζL(s) has at most one zero ρ = β+iγ with β > 1−(4 log d −1L) and |γ| < (4 log d )−1L . Habiba Kadiri [13, Theorem 1.1] in 2012 proved that for sufficiently large dL, ζL(s) has at most one zero ρ = β+ iγ with β ≥ 1− (12.74 log d )−1L and |γ| < 1. But the result that we will use is the one proved by Ahn and Kwon in [2, Theorem 1]. They proved: 68 2.9. FINAL ESTIMATES Theorem 2.37. [2, Theorem 1] If nL ≥ 2, then ζL(s) has at most one zero ρ = β + iγ with β > 1− (2 log dL)−1 and |γ| < (2 log dL)−1 and if this zero exists, then it has to be real and simple. 2.9 Final Estimates We finally conclude by applying Remark 2.31 to estimate ψC(x) and πC(x). 2.9.1 Estimating ψC(x) Lagarias and Odlyzko in [15, Theorem 9.2] proved that there is an effectively computable positive absolute constant c13 such that if x ≥ exp(4n (log d )2L L ), then |C| |C| xβ0 ψC(x) = | |x− | |χ0(g) + ψ(x), (2.125)G G β0 where 1 | ψ(x)| ≤ x exp(− − 1 c 213nL (log x) 2 ), and where the second term on the right side of (2.125) occurs only if ζL(s) has an exceptional zero β0, and χ0 is the (real) character of G0 = Gal(L/E) =< g > for which L(s, χ0) has β0 as a zero. We prove an explicit result to this as Theorem 1.12. √ 25 4 2156 Proof of Theorem 1.12. Let a = e 44 , b = 3 5 e 7−4 3335 and c = 2 × 5 . If ρ = β + iγ with ρ = β0 be the non-trivial zeros of some L(s, χ) with |γ| < T , then by Lemma 2.32 and checking √ ( ( ) ) − 5 log 54 25 −1 c (7 4 3) 22(log dL)+nL (log(|γ|+3))+4c2+ + ≥ , 2 2 2 log((ad )44/5L (b(T + 3))nL) 69 2.9. FINAL ESTIMATES we get ( ( ) ) √ −1 1−(7−4 3) 22(log d 5L)+nL (log(|γ|+3))+4c2+ log 54 + 25 |xρ| 2 2 2= xβ ≤ x ( ( ) ) √ −1−(7−4 3) 22(log dL)+n 5L (log(|γ|+3))+4c + log 542 + 252 2 2 = xx ( ( ) ) √ −1−(log x)(7−4 3) 22(log d )+n 5 (log(|γ|+3))+4c + log 54 + 25L L 2 2 2 2 = xe − c(log x)≤ xe log((ad 44/5 nL) (b(T+3)) L ) , (2.126) ≥ ≥ ≥ ∑ ∑for x 2 and T T0 2. Also, by Lemma 2.16, χ logA(χ) = log dL and χ nE = nL, we get, ∑ ∑ ∣∣∣ ∣ ( )∣1 ∣∣∣ ∑ ∑≤ nχ(2j + 2) + nχ(−(2j + 2)) + 2nχ(0)ρ − 2j + 1χ |ρ|≥ 1 χ2 0≤j≤T 1|γ|≤T ( 2 ( )) ∑( ∑ c3 log(A(χ)) + nE log(2j + 5) + c4 ( ( ))) ≤ + c3 log(A(χ)) + nE (log 3) + c4 2j + 1 χ( 0≤j≤T−12 ( ))( ∑ ) ≤ 1c3 log dL + nL (log(T + 4)) + c4 + 1 . (2.127) 2j + 1 0≤j≤T−1 2 ∑ Recall that log(K + 1) < 11≤j≤K j ≤ (logK) + 1. Therefore ∑T−12 ∑ TT  2  ( (⌊ ⌋ ))1 1 − 1 ∑ 1= ≤ (log T ) + 1− 1 Tlog + 1 2j + 1 j 2 j 0 1 1 ( ( )) 2 2 ≤ 1 T log T(log T ) + 1− log ≤ + 1.35. (2.128) 2 2 2 Therefore using (2.128) in (2.127), we get ∑ ∑ ∣∣∣ ∣∣∣ ( )( ( ))∣1 ∣ ≤ log Tc3 + 2.35 log dL + nL (log(T + 4)) + c4 . (2.129)ρ 2 χ |ρ|≥ 1 |γ|≤2T ∏ Now for ρ = 1 − β0, using ζL(s) = χ L(s, χ) and the result of Ahn and Kwon as given in 70 2.9. FINAL ESTIMATES T∑heorem 2.37, we get |γ| ≥ 1 ∑2(log d ) and hence |ρ| ≥ 1 2(log d ) . We use this result, Lemma 2.16,L L χ logA(χ) = log dL and χ nE = nL to obtain: ∑ ∑ (∣∣∣ ∣ xρ ∣∣∣ ∣∣ ∣∣∣ ∣ 1 ∣∣) √ ∑ ∑≤ ∣∣ ∣1 ∣+ ( x+ 1) ∣ ∣ ρ ρ ∣ ∣ρ ∣ χ ρ=1−β0 χ ρ=1−β0 |ρ|< 1 |ρ|< 1 2 2 √ ∑ ∑≤ ( x+ 1)(2 log dL) 1 χ ρ= 1−β0 ∑ |ρ|< 1 2 ≤ √( x+ 1)(2 log dL) nχ(0) χ √ ∑( ( ( )))≤ c3( x+ 1)(2 log dL) log(A(χ)) + nE (log 3) + c4 ( 2χ( ( ))) ≤ √( x+ 1)(log dL) c(3 (log dL) + n(L (log 3) + c)4√ ) = c3(log dL)( x+ 1) (log dL) + nL (log 3) + c4 . (2.130) Now by Hermite-Minkowski identity, we have n ≤ log√dL = 2 log dLL log 3 for nL > 1. Using this inlog 3 (2.130), we get ∑ ∑ (∣∣∣ ∣∣∣ ∣∣∣ ∣∣∣) √ ( ( ))∣xρ ∣ ∣1+ ∣ ≤ log dLc3(log dL)( x+ 1) (log dL) + √ (log 3) + c4ρ ρ χ ρ − log 3=1 β0 |ρ|< 1 2 √ ( )(log 3) + c4 = c3(log d ) 2 L ( x+ 1) 1 + √ . (2.131) log 3 Note that if nL = 1 (and log dL = 0), then (2.131) is trivially true since ζL has no non-trivial zeros ρ = β+ iγ with |γ| < 14 as mentioned in Remark 2.35. We remember that 0 < 1−β 10 ≤ 2 . Therefore by Mean Value Theorem, x1−β0 1−β0 − − 1 x − 1 √ − = Ω 1−β0 1 β0 1 β0 1− ≤ max x (log x) ≤ x (log x) ≤ x(log x). (2.132)β0 0<Ω≤1−β0 71 2.9. FINAL ESTIMATES Now using (2.100), we have ∣∣∣∣ |C| xβ0− ∣ ∣∣ | | ∣∣∣∑ (∣ C ∣ ∑ xρ ∑ xρ ∑ ) ∣ − 1 − x β0 ∣ S(x, T ) | |χ0(g) = | | χ(g) + χ0(g) ∣∣. (2.133)G β0 G ρ ρ ρ βχ | |≥ 1 | | 1 |ρ|< 1 0ρ ρ < | 2 2 2γ| 1, then ζL(s) has at 79 2.10. REASONS FOR IMPROVEMENTS TO WINCKLER’S RESULTS AND THEIR IMPACT most one zero ρ = β + iγ in the region |γ| ≤ (4 log d −1L) and β ≥ 1− (4 log d −1L) . Instead, we use the result proved by Ahn and Kwon and shown in Theorem 2.37 which gives us the zero-free region |γ| ≤ (2 log dL)−1 and β ≥ 1− (2 log dL)−1, which is clearly an improvement to the one Winckler used. Additionally, Winckler uses the Hermite-Minkowski inequality n ≤ log dL whereas we use n ≤ 2 log dLL log π L log 3 instead. All 3 these changes contribute towards the improvements we obtain. 3. For computing the bound for the error term Eψ(x), Winckler u(sed x ≥ 2, which is a trivia)l 44 lower bound on x. But we notice that the condition x ≥ exp 4nL(log(1 114 759 d 5L ))2 given in Theorem 1.12 fo(r Eψ(x) ensures that log xn ≥) 2226 and thus log x ≥ 4452. Similarly,L 44 the condition x ≥ exp 8n (log(1 114 759 d 5 ))2L L given in Theorem 1.10 ensures that log x n ≥ 4452 and thus log x ≥ 2×4452. Since the bounds for the error terms are decreasingL in x, large values of x will give improved results. This is the main factor which makes such a significant change in our values than that of Winckler’s. Table 2.2: Improving Winckler’s results Results from Winckler [30] Results in this thesis Theorem 1.10 Theorem 1.1 Theorem 1.11 Lemma 5.1 Lemma 2.19 Theorem 8.2 Theorem 1.12 80 Chapter 3 A new explicit version of Chebotarev’s density theorem 3.1 Introduction The objective of this chapter is to provide a new explicit version of Lagarias and Odlyzko’s result on Chebotarev’s density theorem. We derive an asymptotic formula with an explicit error term for a weighted prime power counting function ∑ ψC(x) = log(N p). N pm≤x p unramified σmp =C Our main objective going ahead will be to study the error term Eψ defined by ∣ |C| ∣ ∣∣∣ψC(x)− |G|x ∣Eψ(x) = |C| ∣∣. (3.1) |G|x Recall that dL denotes the absolute discriminant of the number field L and nL = [L : Q]. In this chapter, we prove two main theorems based on the dependency of Eψ(x) on dL. The first one has Eψ(x) dependent on dL : Theorem 1.14. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let β0 be the possible exceptional real zero of ζL(s). Let m ≥ 2 be an integer. Let R = 29.57. If 1/n log x ≥ 4mRnL(log 88d LL )2, then xβ0−1 Eψ(x) ≤ + 1(m,x, nL, dL), β0 81 3.1. INTRODUCTION with { } ( 1 √ ) − 1 m 1 2 (m,x, n , d ) = λ(m)max (log d )n m+11 L L L , Dm+1 (log x)m+1L exp − 2m log x (3.2) m+ 1 RnL where λ is defined in (3.163). Corollary 3.2. Under the assumptions in Theorem 1.14, we have xβ0− ( √ ) 1 ≤ − 1 22 3n 1 log x Eψ(x) +A0max{(log dL)n 3L , 2 3d L }(log x) 3L exp −B0β0 nL for all ≥ (log d 2 L) log x C0 nL where A0 = 0.782 if β0 exists and 0.493 otherwise, B0 = 0.173 and C0 = 19 810 . The second theorem has Eψ(x) independent of dL: Theorem 1.15. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let β0 be the possible exceptional real zero of ζL(s). Let m ≥ 2 be an integer. Let R = 29.57. If 1/n log x ≥ 4mRnL(log 88d L)2L , then xβ0−1 Eψ(x) ≤ + 2(m,x, nL), β0 with ( 1 √ ) 1− 1 1 1.5m 2 log x 2(m,x, n ) = ν(m) n m+1 L L (log x)m+1 exp − (3.3)m+ 1 RnL where ν is defined in (3.167). Corollary 1.16. Under the assumptions in Theorem 1.15, we have ( √ ) xβ0−1 2≤ 1 log x (log d ) 2 L Eψ(x) +A1n 3 L(log x) 3 exp −B1 for all log x ≥ C1β0 nL nL where A1 = 0.0396 if β0 exists and 0.0249 otherwise, B1 = 0.13 and C1 = 19 810 . 82 3.1. INTRODUCTION Corollary 3.5. Under the assumptions in Theorem 1.15, we have xβ0− ( √ ) 1 ≤ log xEψ(x) +A2 exp −B2 for all log x ≥ C2nL(log d 2L) β0 nL where A2 = 3.11, B2 = 0.125 and C2 = 4953. Remark 3.6. Another admissible value for (A2, B2, C2) is (1.84×10−4, 0.014, 4953). Winckler in [30, Theorem 8.2] proved (A2, B2, C2) = (1.51× 1012, 0.014, 1545). Theorem 1.17. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let β0 be the possible exceptional real zero of ζL(s). Then ∣∣∣ ∣ ( √ )∣ |C| ∣ |C| |C| log xπC(x)− | | Li(x)∣∣ ≤ | | Li(xβ0) + E0 | |nLx exp − F0G G G nL for all ≥ (log d 2 L) log x G0 nL where E0 = 4.714× 10−6 if β0 exists and 2.97× 10−6 otherwise, F0 = 0.0919 and G0 = 39 620. Corollary 1.18. Under the assumptions in Theorem 1.17, we have ∣∣∣ | | ∣∣∣ | | | | ( √ )∣ − C ∣ ≤ C β C log xπC(x) | | Li(x) | | Li(x 0) + E1 | |x exp − F1G G G nL for all log x ≥ G1nL(log dL)2 where E1 = 1.65× 10−5, F1 = 0.09 and G1 = 9906. Remark 3.9. Another admissible value for (E1, F1, G1) is (1.23× 10−9, 1/99, 9 906). Winckler in [30, Theorem 1.1] proved (E1, F1, G1) = (7.84× 1014, 1/99, 3 090 ). Next, we give some pre-requisites concerning the zeros of the Dedekind ζ-function, ζL(s). Theorem 3.10. ([11, Corollary 1.2]). Let T ≥ 1 and NL(T ) be the number of zeros ρ = β+ iγ of ζL(s) in the region 0 < β < 1 and |γ| ≤ T . Then ∣∣∣∣ − ∣ ∣ NL(T ) P (T )∣∣ ≤ E(T ), (3.4) 83 3.2. INTRODUCING A SMOOTH WEIGHT where ( ( ) ) T T nL P (T ) = log dL and E(T ) = α1(log dL + nL log T ) + α2nL + α3, π 2πe with α1 = 0.228, α2 = 23.108 and α3 = 4.520. (3.5) Theorem 3.11. Let L be a number field with nL ≥ 2. Then there exists α4 > 0 such that the Dedekind zeta function, ζL(s) has at most one zero ρ = β + iγ with − 1β > 1 and |γ| 1< . (3.6) α4 log dL α4 log dL This zero, if it exists, has to be real and simple, and denoted as β0. Table 3.1: Improvements to α4 Author Region α4 Stark [27, Lemma 3] All dL 4 Kadiri [12, Corollary 1.2] dL sufficiently large 2 Ahn and Kwon [2, Theorem 1] All dL 2 In this thesis, we will use α4 = 2. (3.7) Theorem 3.12. Let L be a number field with nL ≥ 2. Let ρ = β + iγ be non-trivial zero of ζL(s) with ρ = β0 and τ = |γ|+ 2. Then there exists R > 0 such that 1− β > (RL log(Dτ/2))−1, (3.8) 1 n where RL = Rn LL, D = 2dL . Ahn and Kwon in [3, Proposition 6.1] proved that R = 29.57. (3.9) 84 3.2. INTRODUCING A SMOOTH WEIGHT 3.2 Introducing a smooth weight Let 0 < δ ≤ 1, α = 1− δ or 1, m ∈ N and m ≥ 2. We define a function h on [0,∞) by ⎪⎧⎪⎪⎪⎪⎪⎨ 1 if 0 ≤ x ≤ α, h(x) = ⎪⎪⎪⎪⎪⎪ g(x−αδ ) if α ≤ x ≤ α+ δ, (3.10) ⎩ 0 if x ≥ α+ δ, where g is a function defined on [0, 1] satisfying 1. (Condition 1) 0 ≤ g(x) ≤ 1 for 0 ≤ x ≤ 1, 2. (Condition 2) g is an m-times differentiable function on (0, 1) such that for all k = 1, ...,m, g(k)(0) = g(k)(1) = 0, and there exist positive constants ak such that |g(k)(x)| ≤ ak for all 0 < x < 1. The Mellin transform of h is given by ∫ ∞ H(s) = h(t)ts−1dt. (3.11) 0 Also if H is analytic in (s) > 0, the inverse Mellin transform formula is given by ∫ 1 2+i∞ h(t) = H(s)t−sds. 2πi 2−i∞ By the definition of h and g as above, we can check that ∫ α+δ | 1h(m+1)(t)|tm+1dt = M(δ,m), α δ m 85 3.2. INTRODUCING A SMOOTH WEIGHT where for any non-negative integer m, we define M(δ,m) = max M(α, α+ δ,m) α=1−δ, 1 with ∫ 1 M(α, α+ δ,m) = |g(m+1)(u)|(δu+ α)m+1du. (3.12) 0 Let 0 < δ ≤ 0.01, α = 1− δ or 1, m ∈ N and m ≥ 2. [8, Lemma 2.2] gives us 1. The Mellin transform H of h has a single pole at s = 0 with residue 1 and is analytic everywhere else. 2. Let s ∈ C such that (s) ≤ 1. Then H satisfies ∫ 1 H(1) = α+ δ g(u)du, (3.13) 0 | | ≤ M(α, α+ δ, k)H(s) | | , for all k = 0, 1, ...,m. (3.14)δk s k+1 3.2.1 Choice of weight g(x) In this subsection, we define two choices of smooth weights, i.e, g(x) = gR(x) or gFK(x). 1. The first choice of smooth weight, gR is such that it is equivalent to the approach of Rosser in [23] and is demonstrated in [8, Section 3.4] giving the smooth function ∑m ( )( ) ( )1 − j+m m (1 + (δ + 2(α− 1))j/m)− x m xhR(x) = ( 1) , m! j δ/m 1 + (δ + 2(α− 1))j/m j=0 (3.15) where is the indicator function on (0, 1), with its Mellin transform ∑m ( )j+m+1 m j=0(−1) j (1 + (δ + 2(α− 1))j/m)m+s HR(s) = . (3.16) (δ/m)ms(s+ 1) · · · (s+m) From (3.15), we deduce that ∑m ( ) ( ) ∫1 m x α+δ δ gR(x) = (−1)j+m (j − x)m with hR(x)(x) dt = , (3.17) m! j j j=0 α 2 86 3.3. INTRODUCING SMOOTHED VERSION OF ψC(X) and using [23, Theorem 15], we obtain ⎪⎧⎪⎨ (m((1 + δ/m)m+1 + 1))m for m ≥ 1, and MR(δ,m) = ⎩⎪⎪ (3.18)1 + δ2 for m = 0. 2. The second choice of smooth weight is as chosen by Faber and Kadiri [8], that is ∫ (2m+ 1)! x gFK(x) = 1− tm(1− t)mdt, (3.19) (m!)2 0 and some associated properties are ∫ 1 0 gFK(u)du = 1 2 , MFK(α, α+ δ, 0) = 2α+δ 2 , M (δ, 0) = 1 + δ ≤ 2.01FK 2 2 since δ ≤ 0.01, and MFK(α, α+δ,m) for this choice of smooth weight is bounded as shown in [8, Equation 3.5] with √ √ (α+ δ)2m+3 − α2m+3 (2m)!(2m+ 1)! MFK(α, α+ δ,m) ≤ . δ(2m+ 3) m! For 1− δ ≤ α ≤ 1, we obtain √ √ (1 + δ)2m+3 − 1 (2m)!(2m+ 1)! MFK(δ,m) = . (3.20) δ(2m+ 3) m! √ (2m)!(2m+1)! We check√that MR(δ,m) ≈ (2m) m and MFK(δ,m) ≈ m! as δ is close to 0. Also m (2m)!(2m+1)!(2m) < m! as soon as m ≤ 5. We also check that for 0 < δ ≤ 10−100, MR(δ,m) ≤ MFK(δ,m) for m ≤ 5 and MR(δ,m) > MFK(δ,m) otherwise. In this chapter, we use only the first choice of smooth weight, since it produces better bounds for smaller m values, in particular, m = 2. 87 3.3. INTRODUCING SMOOTHED VERSION OF ψC(X) 3.3 Introducing smoothed version of ψC(x) We introduce the smooth version of ψC(x) as ∑ ∑ ( )N pm ψ̃C(x) = (log(N p))h , (3.21) x p unramified m≥1 σmp =C where p runs over all the prime ideals of K and h is defined in (3.10). Recall our definition of Eψ in (3.1). Similarly Eψ̃ denote the error term corresponding to ψ̃C(x) and defined as ∣ ∣∣∣ ∣ ψ̃C(x)− |C||G|x ∣ Eψ̃(x) = | | ∣∣. (3.22)C |G|x We define ψ−C and ψ + C as the sums ψ̃C associated to the weights h defined by α = 1 − δ and α = 1 respectively. We also denote E−ψ and E + ψ the respective error terms. Observe that ψ− +C (x) ≤ ψC(x) ≤ ψC (x), (3.23) and Eψ(x) ≤ max(E− +ψ (x), Eψ (x)). (3.24) We write ψ̃C(x) = IL/K(x)− ĨL/K(x), (3.25) with ∑ ∑ ( )N pm IL/K(x) = θ(p m)(log(N p))h , (3.26) x p m≥1 and ∑ ∑ ( )N pm ĨL/K(x) = θ(p m)(log(N p))h , (3.27) x p ramifiedm≥1 where θ is the indicator function characterizing the Artin symbol at p coinciding with the conjugacy class C. More specifically, for p unramified in L, we have ⎪⎪⎧⎨ 1 if σmp = C, θ(pm) = ⎪⎪⎩ (3.28)0 otherwise, 88 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) and |θ(pm)| ≤ 1 if p ramifies in L. 3.4 Controlling the smoothed sum over ramified prime ideals Lemma 3.13. Let C be a fixed conjugacy class of G = Gal(L/K). For x ≥ x0 ≥ exp(4484), 0 ≤ δ ≤ δ0 ≤ 10−10 and α = 1− δ or 1, we have ∣∣∣ ∣∣ ĨL/K(x) ∣|C| ∣∣ ≤ log x0(log dL) | |x xG where 0 depends on x0 and δ0, and is given by ( ) 2 log(1 + δ0) 0 = 1 + ≤ 2.89. (3.29) log 2 log x0 Proof. By definition of θ in (3.28) and h in (3.10), we get ∑ ∑ ∑ ∑ |ĨL/K(x)| ≤ (log(N p)) ≤ log(N p) 1. (3.30) p ramified m≥1 p ramified m≥1 N pm be the cyclic group generated by g, E be the fixed field of G0, and χ denotes the irreducible characters of G0. Let H be defined in (3.11). Then | | ∑ ( ∫ 2+i∞ ( ′ ) )C 1 L IL/K(x) = | | χ(g) H(s)x s − (s, χ, L/E) ds . G 2πi 2−i∞ Lχ Proof. Let φ be a irreducible characters of G = Gal(L/K). Recall from (2.2) that ∑ φ (pm 1 m K ) = | | φ(τ α),I0 α∈I0 where I0 is the inertia group of q, one of the prime ideal factors of p, and τ is one of the Frobenius automorphism corresponding to p. If L(s, φ, L/K) is the Artin L-series associated to φ, then from [15, (3.2)], we get that for (s) > 1, −L ′ ∑ (s, φ, L/K) = φ (pmK ) log(N p)(N p) −ms, L p,m where the outer sum is over all the prime ideals of K. Using [15, (3.1),(3.2),(3.5),(3.6)], θ defined in (3.28) can be redefined in a general way as ∑ θ(pm |C| ) = | || | φ(g)φ(τ mα). (3.34) I0 G φ α∈I0 Using (3.34) and the inverse Mellin transform of h in (3.26), we obtain |C| ∑ ( ∫ 2+i∞ ( ′ ) )1 L IL/K(x) = | | φ(g) H(s)x s − (s, φ, L/K) ds . G ∈ 2πi 2−i∞ Lφ G 90 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) Deuring reduction as shown in [15, Lemma 4.1] is the process of reduction of Artin L-functions, L(s, φ, L/K) to the case of Hecke L-functions, L(s, χ, L/E) where intermediate field extension L/E has a cyclic Galois group. Hecke L-functions have been proven to be holomorphic on the entire complex plane whereas th∑e same has no∑t been proven for the Artin L-functions. Following Deuring reduction, we obtain, χ χ(g)χ ∗ = φ φ(g)φ where χ ∗ is the character of G induced by G0. Also [18, Theorem 2.3.2(d)] gives us L(s, χ ∗, L/K) = L(s, χ, L/E). Therefore, |C| ∑ ( ∫1 2+i∞ ( ′ ) )L IL/K(x) = | | χ(g) H(s)x s − (s, χ, L/E) ds , G 2πi 2−i∞ Lχ which holds for (s) > 1, and hence by analytic continuation, for all s. 3.5.2 Obtaining formula for IL/K From now on, we will denote L(s, χ) for L(s, χ, L/E). We let F (χ) denote the conductor of χ and set A(χ) = dENE/Q(F (χ)). Let δ(χ) be defined as ⎪⎧⎨⎪ 1 if χ = χ1, the principal character, δ(χ) = ⎩⎪⎪ (3.35)0 otherwise. From (2.44) - (2.48), we recall that, for each χ, there exist non-negative integers a(χ) and b(χ) such that a(χ)+ b(χ) = nE , and such that if we define γχ(s) as in (2.46) and ξ(s, χ) as in (2.47), then ξ(s, χ) satisfies the functional equation (2.48). From (2.52), we recall that, for all complex number s, ′ ∑( ) ( )L 1 1 1 1 1 γ′χ (s, χ) = B(χ) + − + − δ(χ) + − − log(A(χ))− (s), (3.36)L s ρ ρ s s 1 2 γ ρ χ where B(χ) is a constant which depends on χ but not defined explicitly and ρ denotes the non-trivial zeros of L(s, χ), i.e., ρ = β + iγ with 0 < β < 1. Theorem 2.12 showed that ∑ ( )  1B(χ) = −  . (3.37) ρ ρ 91 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) Also, from (2.94), recall that, r(χ) is a constant defined as ( ) − 1 nE b(χ) Γ ′ 1 a(χ) Γ′ r(χ) = B(χ) (logA(χ)) + (log π) + δ(χ)− − (1). (3.38) 2 2 2 Γ 2 2 Γ Lemma 3.15. Let s = σ + it and ρ = β + iγ denotes a zero of L(s, χ) with 0 < β < 1. 1. For σ ≤ −1/4 and |s+m| ≥ 1/4 for all non-negative integer m, L′ (s, χ)  logA(χ) + nE log(|s|+ 2). (3.39) L 2. For −1/2 ≤ σ ≤ 3 and |s| ≥ 1/8, ∣∣∣∣L′ ∣ δ(χ) ∑ (s, χ) + − − 1 ∣∣∣−  logA(χ) + nE log(|t|+ 2). (3.40)L s 1 s ρ ρ |γ−t|≤1 3. Suppose nχ(t) = #{ρ | L(ρ, χ) = 0, |γ − t| ≤ 1}. Then for all t, nχ(t)  logA(χ) + nE log(|t|+ 2). (3.41) 4. Suppose γ = t, |t| ≥ 2, x ≥ 2 and 1 < σ1 ≤ 3, then ∫ σ1 xσ+it − dσ  |t| −1xσ1(σ1 − β)−1. (3.42) −1/4 (σ + it)(σ + it ρ) Proof. See [15, Lemma 6.2, Lemma 5.6, Lemma 5.4, Lemma 6.3]. Proposition 3.16. Let C be a conjugacy class of G = Gal(L/K), g ∈ C, G0 =< g > be the cyclic group generated by g, E be the fixed field of G0, and χ runs through the irreducible characters of G0. Let 0 < δ ≤ 0.01, α = 1− δ or 1, h be defined in (3.10) and H be defined in (3.11). Then for x ≥ 2, ( | | ∑ ( ∫C α+δ )h(t) IL/K = xH(1) + | | χ(g) − r(χ)− (a(χ)− δ(χ)) (logαx) + dtG α t ∑ χ ∑ ∑ ) − xρH(ρ)− b(χ) x−(2m−1)H(1− 2m)− a(χ) x−2mH(−2m) , (3.43) ρ∈Z(χ) m≥1 m≥1 92 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) where Z(χ) denote the set of non-trivial zeros of L(s, χ), which are precisely those zeros ρ = β+iγ of L(s, χ) for which 0 < β < 1. Proof. Let x ≥ 2 and T ≥ 2 be such that it does not equal the ordinate of any zero of any of the L(s, χ). We rewrite (3.5.1) as | | ∑ ( )C IL/K(x) = |G| χ(g) lim Jχ(x, T ) , (3.44)T→∞ χ where ∫ 1 2+iT ( ) L′ Jχ(x, T ) = Yχ(s)ds with Yχ(s) = H(s)x s (s, χ) . (3.45) 2πi 2−iT L Let U = j + 12 for some non-negative integer j and BT,U be the positively oriented rectangle with vertices at 2− iT , 2 + iT , −U + iT and −U − iT . We define ∫ 1 Jχ(x, T, U) = Yχ(s)ds. (3.46) 2πi BT,U The next step is to study the poles for Yχ(s), followed by using Cauchy’s theorem and then take the limit as T, U → ∞. The poles of Yχ(s) inside BT,U and their contributions are: ′ 1. At s = 0. The Laurent series expansion of LL (s, χ) about s = 0 as defined in [15, Page 448, (7.1)] shows that L′ a(χ)− δ(χ) (s, χ) = + r(χ) + sf(s, χ), L s where f(s, χ) is a function that is analytic at s = 0. Also, Kadiri and Lumley in [14, (3.3),(3.4)] showed that H(s)xs has a simple pole at s = 0 and its Laurent series expansion at this point is ∫ 1 α+δs h(t)H(s)x = (1 + (log x+G′(0))s+O(s2)) with G′(0) = logα+ dt. (3.47) s α t Therefore the residue of Yχ(s) at s = 0 is ( ∫ α+δ ) −r(χ)− (a(χ)− h(t)δ(χ)) (logαx) + dt . α t 93 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) 2. At s = 1. We know that H(s)xs is analytic at s = 1, as well as L(s, χ) unless χ = χ1, the ′ principal character, in which case LL (s, χ) has a first order pole of residue −1 at s = 1. Hence, the residue of Yχ(s) at s = 1 is δ(χ)xH(1). ′ 3. At non-trivial zeros of L(s, χ). LL (s, χ) has a first order pole with residues +1 at each non-trivial zero ρ of L(s, χ) (counted with multiplicity). Also F (s)xs is analytic at such points. Hence, the residue of Yχ(s) at Z(χ) is ∑ − xρH(ρ). ρ∈Z(χ) |γ| 0, lim V1,χ(x, T, U) + V2,χ(x, T, U) + V→∞ 3,χ (x, T ) = 0. (3.58) T,U ∑ Combining (3.44), (3.48), (3.58) and χ δ(χ) = 1, we obtain the announced identity (3.43). ∏ Now since χ L(s, χ) = ζL(s), therefore any sum over the non-trivial zeros of all the L(s, χ)’s 96 3.5. EXPLICIT FORMULA FOR SMOOTHED SUM OVER ALL PRIME IDEALS (IL/K) can be considered as a sum over the non-trivial zeros of ζL(s). Thus rewriting ∑ ∑ 1 ∑ 1 ∑ r(χ) + xρH(ρ) = r(χ) + − + xρH(ρ) ρ ρ ρ∈Z(χ) ρ∈Z(χ) ρ∈Z(χ) ρ∈Z(χ) |ρ|< 1 |ρ|< 1 2 2 ∑ ∑ and using χ |a(χ)− δ(χ)| ≤ χ nE = nL in (3.43), we obtain : Corollary 3.17. Let 0 < δ ≤ 0.01, α = 1− δ or 1, h be defined in (3.10) and H be defined in (3.11). Then for x ≥ 2 and T ≥ 2, ∣∣∣ | | ∣∣∣ (| | ∣∣ − C ∣ ≤ C ∣ ∫ ∣α+δ h(t) ∣ I xH(1) xβ0H(β ) + n ∣∣(logαx) + dt∣∣+ J (0)(x) + J (1)(x) + J (2)L/K |G| |G| 0 L (x)( α t )) + J (3)(x) + x J (4)(x) + J (5)(x, T ) + J (6)(x, T ) , (3.59) where ∑( ∑ ∑ ) J (0)(x) = b(χ) x−(2m−1)|H(1− 2m)|+ a(χ) x−2m|H(−2m)| , (3.60) ∑χ ∣∣ m≥∑1 ∣∣ m≥11 J (1) = ∣∣r(χ) + ∣∣, (3.61)ρ χ |ρ|< 1 2 J (2)(x) = x1−β0H(1− 1∣β0)−∑ ∣ 1− , (3.62) β0∣∣ J (3) 1 (x) = ∣∣xρH(ρ)− ∣∣, (3.63)ρ ρ =1−β0,|ρ∑|< 1 2 J (4)(x) = xβ−1|H(ρ)|, (3.64) ρ =β0,|ρ|≥ 1∑,|γ|≤ 1 2 α4 log dL J (5)(x, T ) = xβ−1|H(ρ)|, (3.65) |ρ∑|≥ 1 , 1 <|γ| 0 in this section. This is achieved by generalizing a new technique employed for Dirichlet L-functions by Fiorilli and Martin in [10, Section 5]. In [10, Lemma 5.3], they used 2+ iT in their sum to obtain their result. We, on the other hand, use 1 + + iT instead to get closer to the (s) = 1 line and also provide a general result depended on both a and . First, from [30, Lemma 4.3, Lemma 4.5], we recall that 98 ∣∣ ∣ 3.6. EXPRESSING ∣∣I − | ∣C| ∣L/K |G|XH(1)∣ IN TERMS OF SUM OVER ZEROS Lemma 3.18. If (s) > 1, then ∣∣∣∣L′ (s, χ)∣ ∣∣ L ∣ ≤ nE . (3.69)(s)− 1 and if (s) > −12 and |s| ≥ 18 , then ∣∣∣∣γ′χ (s)∣ ∣∣ ( )∣ ≤ nE log(1 + | 164s|) + . (3.70)γχ 2 7 Lemma 3.19. Let 0 < ≤ 1 and let ρ = β + iγ be the non-trivial zero of L(s, χ). For any real number T , ∑ ( ( ))1 1 1 164 logA(χ) 2δ(χ) | − | < nE log(2 + + |T |) + + + + .∈ 1 + + iT ρ 2 2 2 14 2 ρ Z(χ) ′ Proof. By the classical explicit formula for LL (s, χ) given in (2.52) and (3.37), we notice that ( ∑ ) ( )  1  L ′ logA(χ) − = (1 + + iT, χ) +∈ 1 + + iT ρ Lρ Z(χ) ( ( 2)) ( 1 1 γ′ ) χ + δ(χ)  + +  (1 + + iT ) . 1 + + iT + iT γχ Using Lemma 3.18 in the above equation, we obtain ( ∑ ) ( ( ))  1 1 1 164 logA(χ)< nE log(2 + + |T |) + + + + 2δ(χ). (3.71) ρ∈ 1 + + iT − ρ 2 14 2Z(χ) ∑ (∑ ) Using this with the fact that 1 1 1ρ∈Z(χ) |1++iT−ρ|2 <  ρ∈Z(χ) 1++iT−ρ , we obtain the required result. Proof of Proposition 1.13. Since 0 < β < 1, therefore < 1 + − β < 1 + . Using this we 99 ∣∣ ∣∣ 3.6. EXPRESSING ∣∣I − |C|| |XH(1)∣L/K G ∣ IN TERMS OF SUM OVER ZEROS get ∑ ≤ (1 + ) 2 + a2 ∑ 1 + − β 1 (1 + − β)2 + (T − γ)2 ρ∈Z(χ) ρ∈Z(χ) |T−γ|≤a ∑ ( )(1 + )2 + a2 1 =  − . ∈ 1 + + iT ρρ Z(χ) Thus using (3.71), we obtain the required result. Lemma 3.20. Let 0 < ≤ 1. If s = σ + iT with −12 < σ ≤ 2 and |s| ≥ 18 , then ∣∣∣∣L′ ∣ δ(χ) ∑ 1 ∣ (s, χ) + ∣− − − ∣ ≤ nE(c5(a, ) log(3 + |T |) + c6(a, )) + (c5(a, )− 1) logA(χ)L s 1 s ρ ρ∈Z(χ) |γ−T |≤a + c7(a, )δ(χ), where ( )√ ( ) 1 3 1 + 2 c3(a, ) = + 1 + , (3.72) (2 2 )√ ( a) 3 1 + 2 ( ) 1 164 c4(a, ) = + 1 + + , (3.73) 2 a 2 14 c1(a, ) c5(a, ) = + c3(a, ) + 1, (3.74) 1 c2(a, ) 164 c6(a, ) = + + + c4(a, ), (3.75) 7 4c1(a, ) c7(a, ) = + 4c3(a, ) + 10. (3.76) ′ ′ ′ Proof. Using the formula for LL (s, χ) as in (2.52) and studying the difference L L L (s, χ)− L (1 + 100 ∣∣ ∣∣ 3.6. EXPRESSING ∣∣I |C| ∣L/K − |G|XH(1)∣ IN TERMS OF SUM OVER ZEROS + iT, χ), we obtain ∣∣∣∣L′ ∣ δ(χ) ∑− 1 ∣(s, χ) + ∣ L s− 1 s− ρ ∣ ρ∈Z(χ) ∣∣ |γ∣∣−T |∣∣≤a∣ ′ ′ ′ ∣ ∣ ∣≤ ∣L γ γ ∣ ∑χ χ ∣ 1 1 ∣(1 + + iT, χ)∣L ∣+ ∣∣ (1 + + iT )− (s)∣∣+γ γ ∣∣ ∣χ χ s− −ρ 1 + + iT − ρ ∣ ∣ ρ∈Z(χ)∑ ∣ |γ−T |>a∣1 1 1 1 ∣ + | − | + δ(χ)∣∣ + − ∣. (3.77)1 + + iT ρ 1 + + iT + iT s ∣ ρ∈Z(χ) |γ−T |≤a We bound the terms on the right side individually. By Lemma 2.13, we obtain ∣∣∣ ∣∣L′ ∣ nE(1 + + iT, χ)∣ ≤ .L ∣ By Lemma 2.15 and using ≤ 1, we obtain ∣∣∣ ′ ′ ∣∣ ( )∣γχ γχ 164(1 + + iT )− (s)∣∣ ≤ nE log(3 + |T |) + .γχ γχ 7 Remember that −12 < σ < 2, thus 1 + − σ ≤ + 32 . Using this, we get ∑ ∣ ∣∣∣∣ 1 − 1 ∣∣∣ ∑ 1 + − σs− ρ 1 + + iT − =ρ |s− ρ||1 + + iT − ρ| ρ∈Z(χ) ρ∈Z(χ) |γ−T |>a (|γ−T |>a) 3 ∑ |1 + + iT − ρ| 1 < + 2 |s− ρ| |1 + + iT − .ρ|2 ρ∈Z(χ) |γ−T |>a √ ( )2 Using Lemma 3.19 and the fact that |1++iT−ρ||s−ρ| ≤ 1 + 1+a , we get ∑ ∣∣∣∣ ∣ 1 − − 1 ∣∣ s ρ 1 + + iT − ρ ∣ ρ∈Z(χ) |γ−T |>a < nE(c3(a, ) log(2 + + |T |) + c4(a, )) + c3(a, ) logA(χ) + 4c3(a, )δ(χ). 101 ∣∣∣ ∣3.6. EXPRESSING ∣I − | ∣C|L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS Since 0 < β < 1, therefore, |1 + + iT − β − iγ| > . Using this, we get ∑ 1 ∑ | − | ≤ 1 nχ,a,(T ) 1 = . 1 + + iT ρ ρ∈Z(χ) ρ∈Z(χ) |γ−T |≤a |γ−T |≤a Now using Proposition 1.13 in the above equation, we get ∑ ( )1 ≤ c1(a, ) c2(a, ) c1(a, ) 4c1(a, )| − | nE log(2+ +|T |)+ + logA(χ)+ δ(χ).1 + + iT ρ ρ∈Z(χ) |γ−T |≤a Finally, since |s| ≥ 18 , thus ∣∣∣ ∣∣ 1 1 − 1 ∣δ(χ) + ∣ ≤ 10δ(χ).1 + + iT + iT s ∣ Thus using ≤ 1 in (3.77), we obtain the required result. ′ Recall that B(χ) is the undefined constant in the expression for LL (s, χ) given in (2.52). Lagarias and Odlyzko in [15, Lemma 5.5] proved that, for any with 0 < ≤ 1, we have ∑ 1  logA(χ) + nEB(χ) + , (3.78) ρ ρ∈Z(χ) |ρ|< and Winckler made their result explicit which is shown in Lemma 2.17. We prove : Lemma 3.21. For any ∈ (0, 1], we have ∑∣∣∣ ∣ ( ) ( ) ( )∣ ∑ 1 ∣∣∣ ≤ 0.547 22.205 4.520B(χ) + 3.19 + log dL + 79.251 + nL + 15.06 + .ρ χ ρ∈Z(χ) |ρ|< Proof. Let r be a positive real number. We split the sum over the non-trivial zeros of L(s, χ) as the following ∑ 1 ∑ ( )1 1 ∑ ( )− 1 1 ∑ 1 ∑= + − + − − − − 1 .ρ ρ 1 + r ρ ρ 1 + r ρ 1 + r ρ ρ ρ∈Z(χ) ρ∈Z(χ) ρ∈Z(χ) ρ∈Z(χ) ρ∈Z(χ) |ρ|< |ρ|≥1 |ρ|<1 ≤|ρ|<1 (3.79) 102 ∣∣ 3.6. EXPRESSING ∣ ∣∣I − |C| ∣L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS ∑ Note that the second sum can be compared to the convergent sum 1|ρ|≥1 | |2 . In addition, inρ the last two finite sums, we have, 1 1 1 1|1+r−ρ| < r and |ρ| ≤  . Using (3.79), we obtain ∑∣∣∣∣ ∑ ∣∣ ∣ ∑∣∣∣ ∑ ( )∣ ∣ ( )∣1 ∑ ∑B(χ) + ∣ ≤ ∣ 1 1 ∣ ∣ 1 1 ∣B(χ) + − + ∣∣+ ∣∣ +ρ 1 + r ρ ρ 1 + r − ρ ρ ∣∣ χ ρ∈Z(χ) χ ∣ ρ∈Z(χ) ∣ ∣ ∣ χ ρ∈Z(χ)|ρ|< ∑∣ |ρ|≥1 + ∣∣ ∑ 1 ∣− ∣ ∣ ∑∣ ∑ 1 ∣ + ∣∣ ∣∣. (3.80)1 + r ρ ρ χ ρ∈Z(χ) χ ρ∈Z(χ) |ρ|<1 ≤|ρ|<1 ∏ Let Z(ζ) be the set of zeros of ζL(s), ρ = β +∣∣ iγ with 0∣∣< β < 1. Since χ L(s, χ) = ζL(s), t∣hus Z(ζ) =∣ ∪χZ(χ). For |ρ| ≥ 1/2, we use ∣∣ 1 1− + ∣ 1+r 1+r∣ ∣ 1+r ρ ρ ∣ = |(1+r−ρ)ρ| ≤ | |2 for |γ| > 1, andγ ∣∣ 1 1 ∣ 1+r1+r−ρ + ρ ∣ ≤ r for |γ| ≤ 1 to obtain ∑∣∣∣ ∑ ( )∣∣ 1 1 ∣∣ ∑− + ∣ ≤ 1 + r 1 + rNL(1) +1 + r ρ ρ r |γ|2 χ ρ∈Z(χ) ρ∈Z(ζ) |ρ|≥1 |γ|>1 ∞ ≤ 1 + r ∑ NL(k + 1)−NL(k) NL(1) + (1 + r) 2 ( r ) kk=1 1 ∑∞ = − 2k − 1r NL(1) + (1 + r) NL(k) − . (3.81)r (k(k 1))2 k=2 Using definition of NL(k) as in (3.4), we obtain ( ( )) 1 dL nL NL(k) ≤ log k + (k log k) + αn 1nL log k + α1 log dL + α2nL + α3. (3.82)π (2πe) L π We compute that ∑∞ ≤ k(2k − ∞ 1) ∑≤ k(log k)(2k − ∞1) ∑≤ 2k − 12.645 (k(k − 2.65,1))2 (k(k − 3.06, = 1 and1))2 (k(k − 1))2 k=2 ∑k=2 k=2∞ (log k)(2k − 1) ≤ 0.87. (3.83) (k(k − 1))2 k=2 103 ∣∣ ∣∣ 3.6. EXPRESSING ∣∣I − |C|| |XH(1)∣L/K G ∣ IN TERMS OF SUM OVER ZEROS Note that 1r − r changes sign at r = 1. Thus using (3.83) and (3.82) in (3.81), we obtain ∑∣∣∣∣ ∑ ( )∣ 1 1 ∣ − + ∣ 1 + r ρ ρ ∣ χ ρ∈Z(χ) |ρ|≥{1 }( ( ) ) ≤ 1 1 dLmax 0(, − r log + α1 log dn L + α2nL + α3r π (2πe) L ) 2.65 − 2.645 log(2πe) 3.06+ (1 + r) log dL nL + nL + 0.87α1nL + α1 log dL + α2nL + α3 . π π π (3.84) Note that |1 + r − ρ| > 1 + r − β > r. Again using (3.4), we get ∑∣∣∣ ∣ ∣ ∣ ( )∣ ∑ 1 ∣∣∣ ∑∣ ∑ 1 ∣ 1 1− + ∣1 + r ρ ∣ ∣∣ ≤ + NL(1)ρ r χ ρ∈Z(χ) χ ρ∈Z(χ) |ρ|<1 ≤|ρ|<1 ( )( ( ) ) ≤ 1 1 1 dL+ log + α1 log dL + α nn 2 L + α3 .r π (2πe) L (3.85) For the last part, we use (2.52) with s = 1 + r. Notice that ∑∣∣∣∣ ∑ ( )∣ 1 1 ∣ B(χ) + + ∣∣ χ∑∣ 1 + r − ρ ρ ρ∈Z(χ) = ∣∣ ′ ( ) ′ ∣∣L 1 1 1 γχ ∣(1 + r, χ) + log(A(χ)) + δ(χ) + + (1 + r)∣.L 2 1 + r 1 + r − 1 γ ∣ χ χ ∣∣ ∣∣ ∣∣ ∣′ γ′ ∣ Now( using Lemma 2).13 and Lemma 2.15, we obtain ∣∣LL (1 + r, χ)∣∣ ≤ nE ∣ χ ∣ r and ∣γ (1 + r) ≤χ ∣ nE log(2 + r) + 1642 7 . Thus using (3.68), we have ∑∣∣∣ ∑ ( )∣ ( )∣ 1 1 ∣ 1 1 log(2 + r) 164 2r + 1B(χ) + − + ∣∣ ≤ log dL + + + nL + .∈ 1 + r ρ ρ 2 r 2 14 r(r + 1)χ ρ Z(χ) (3.86) 104 ∣∣ ∣∣ 3.6. EXPRESSING ∣∣I |C| ∣L/K − |G|XH(1)∣ IN TERMS OF SUM OVER ZEROS Finally, we insert (3.84), (3.85) and (3.86) into (3.80) to obtain ∑∣∣∣ ∑ ∣∣∣ ( ) ( ) ( )∣ 1 ∣ ≤ m2 m4 m6B(χ) + m1 + log dL + m3 + nL + m5 + , (3.87)ρ χ ρ∈Z(χ) |ρ|< where { }( ) ( ) 1 1 2.65(1 + r) 1 1 m1 = max 0, − r + α1 + + + 1 + r + α1 + 0.5, r π π rπ r 1 m2 = +{α1,π }( ) ( ) 1 2 − − log(2πe) − 2.645 log(2πe) 3.06m3 = max , r + α2 + (1 + r) + + 0.87α1 + α2 r r π π π 1 log(2 + r) 164 + + + , r 2 14 − log(2πe)m4 = { + α2,π } 1 2 2r + 1 m5 = max 1 + r + , 1 + α3 + , r r r(r + 1) m6 = α3. (3.88) Finally we insert r = 1 and (α1, α2, α3) = (0.228, 23.108, 4.520) as given in Theorem 3.10 into (3.87) to obtain the required result. Remark 3.22. We use the Hermite-Minkowski’s bound, n ≤ log√dLL and check that m + m1 √3log 3 log 3 is minimized at r = 1. Hence we choose r = 1 in the proof of Lemma 3.21. Remark 3.23. Both [15, Lemma 5.5] and [30, Lemma 4.7] used the zero counting function, nχ(t) which denotes the number of non-trivial zeros, ρ = β + iγ of L(s, χ) in the region |γ − t| ≤ 1. We instead use NL(t) defined in (3.4), i.e., the zero-counting function which counts all the zeros, ρ = β +∑iγ of all the characters χ in the region 0 < β < 1 and |γ| ≤ T . We notice that for T ≥ 2, χ(nχ(T ) + nχ(−T )) is equivalent to NL(T + 1) −NL(T − 1). Also, Winckler’s result regarding nχ(T ) in [30, Lemma 4.6] gives ∑ (nχ(T ) + nχ(−T )) ≤ 2.5 log dL + 2.5nL log(T + 3) + 20.06nL, (3.89) χ 105 ∣∣ ∣ 3.6. EXPRESSING ∣∣I − |C| ∣L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS whereas using [11, Corollary 1.2] regarding NL(T ), we obtain NL(T + 1)−NL(T − 1) ≤ 1.093 log dL + 1.093nL log T + 45.11nL + 9.04. (3.90) We use NL(T ) to prove Lemma 3.21 with coefficient of log dL as 3.19 + 0.547  and coefficient of nL as 79.242+ 22.205  . A similar result can be obtained from Lemma 2.17 as proved by Winckler which uses n (T ) to give 10.42 + 1.25 and 101.33 + 11.41χ   as the coefficient of log dL and nL respectively. 3.6.2 Bounding the J (i)’s In this subsection, we bound all J (i)’s defined in Corollary 3.17. Lemma 3.24. Under the assumptions in (3.67),we have ∣∣∣ ∫ ∣∣ α+δ h(t) ∣(logαx) + dt∣∣n + J (0)L (x) + J (1) ≤ 1(log dL)(log x), (3.91) α t where 1 depends on x0 and δ0, and is given by ( ) 4.782 1 δ0 2 + δ0 125.214 25.10 1 = + √ 1 + + + + ≤ 1.842. log x0 log 3 2(1− δ0) log x 2x20 0 log x0 log x0 (log 3)(log x0) (3.92) ∫ α+δ Proof. First of all, using δ ≤ δ0, α = 1− δ or 1, h = hR and α hR(t, α, δ) dt = δ/2, we have ∣∣∣ ∫ ∣∣ α+δ h(t) ∣ δ0(logαx) + dt∣ ≤ log x+ . (3.93) α t ∣ 2(1− δ0) Now using (3.14) for k = 0, we get |H(s)| ≤ M(δ,0)|s| which we insert in (3.60) to obtain ∑( ∑ )x−(2m−1) ∑ −2m J (0) x (x) ≤ M(δ, 0) b(χ) + a(χ) . ≥ 2m− 1 ≥ 2mχ m 1 m 1 ∑ x−(2m−1) ∑ −m ∑ −2mUsing m≥1 2m−1 = x xm≥1 m − m≥1 2m , b(χ) = nE − a(χ) and the Taylor series 106 ∣∣ ∣ 3.6. EXPRESSING ∣∣IL/K − | | ∣C|G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS expansion of log x, we obtain ∑ ( ) ( )x−(2m−1) ∑ x−2m nE x+ 1 x b(χ) − + a(χ) = log − + a(χ) log≥ 2m 1 ≥ 2m 2m 1 m 1 (x 1 ) x+ 1 2 ≤ nE x ≤ nElog . 2 x2 − 1 x2 Thus ∑ (0) ≤ nE ≤ 2 + δ0 nLJ (x) M(δ, 0) . (3.94) x2 2 x2 χ ′ ′ Inserting ΓΓ ( 1 2) = −2(log 2)− γ, ΓΓ (1) = −γ and a(χ) + b(χ) = nE into (2.94), we obtain ∣∣∣∣ ∑ 1 ∣ ∣ ∣∣ ∑ ∣∣ ( )1 logA(χ) log π γ r(χ) + ∣∣ ≤ ∣∣B(χ) + ∣∣+ + δ(χ) + nE + + log 2 .ρ ρ 2 2 2 ρ∈Z(χ) ρ∈Z(χ) |ρ|< 1 |ρ|< 1 2 2 We conclude with Lemma 3.21 applied to = 12 , and with (3.68) : J (1) ≤ 4.782 log dL + 125.214nL + 25.10. (3.95) The announced bound is obtained by putting together (3.93), (3.94), (3.95) and (3.67). Lemma 3.25. Under the assumptions in (3.67), we have J (2) 1 (x) ≤ 2(log x)x 2 , (3.96) where 2 depends on x0 and δ0, and is given by δ0 2 = 1 + √ − ≤ 1.001. (3.97)2 1 δ0 log x0 Proof. Using the equation for H(s) given in (3.11) with h(t) = 1 for 0 ≤ t ≤ α, we have ( ) ∫ α+δ xs 1 1 H(s)− = (αx)s − 1 + xs h(t)ts−1dt. (3.98) s s α 107 ∣∣ 3.6. EXPRESSING ∣ ∣∣I − | ∣C|L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS For s = 1− β ≤ 10 2 , the Mean Value Theorem with α ≤ 1 gives (αx)1−β0 − 1 ≤ x1−β √0− log(αx) ≤ x log x. (3.99)1 β0 ∫ α+δ Since α h(t)dt = δ 2 , thus we get ∫ α+δ √x δ x1−β0 h(t)t−β0dt ≤ √ . (3.100) α α 2 Now combining (3.98), (3.99), (3.100) with the assumptions in (3.67), we obtain the required result. To provide bounds for J (3)(x) defined in (3.63) and J (4)(x) defined in (3.64), we use the zero-free region of ζL(s) as described in Theorem 3.11 with α4 defined in (3.7). Lemma 3.26. Under the assumptions in (3.67), we have J (3)(x) ≤ 13(log d 2L) x 2 , (3.101) where 3 depends on x0 and δ0, and is given by ( )( ( ) ) 2 + δ0 1 1 4c11 3 = α4 + √ c11 + √ c11 log 3 + c21 + ≤ 224.97, (3.102) 2 x0 log 3 log 3 with c 1711 = 8 and c = 1513 21 28 and α4 defined in (3.6). Proof. Notice that using (3.14), we have ∑ ( ≤ ∣∣ ∣(3) xρ ∣J (x) M(δ, 0)∣∣ ∣∣ ∣∣ ∣ ∣) ∣1 ∣∣∣ ∑ ∣ ∣ + ≤ √ ∣1 ∣( xM(δ, 0) + 1) ∣∣ ∣∣. (3.103)ρ ρ ρ ρ∈Z(ζ),ρ= 1−β0 ρ∈Z(ζ),ρ =1−β0 |ρ|< 1 |ρ|< 1 2 2 In the region |ρ| < 12 with ρ = 1 − β0, (3.6) tells us that either |γ| > 1α log d or β > 1α log d ,4 L 4 L which implies that |ρ| > 1α log d . Thus,4 L ∑ ∣∣∣ ∣∣1ρ ∣∣ ∣ ∑ ∑ ∑≤ α4(log dL) 1 ≤ α4(log dL) nχ, 1 ,1(0). (3.104) 2 ρ∈Z(ζ),ρ= 1−β0 χ ρ∈Z(χ),ρ=1−β0 χ |ρ|< 1 |ρ|< 1 2 2 108 ∣∣ ∣ 3.6. EXPRESSING ∣∣I − |C| ∣L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS We apply Proposition 1.13 to bound nχ, 1 ,1. Denoting ci1 instead of ci(1/2, 1), we get 2 ∑ nχ, 1 ,1(0) ≤ (c11 log(3) + c21)nL + c11 log dL + 4c11. (3.105) 2 χ Combining (3.103), (3.104), (3.105) with (3.67) completes the proof. Lemma 3.27. Under the assumptions in (3.67), we have 1 J (4) ≤ −(x)  (log d )x α4 log d4 L L , (3.106) with 4 depends on δ0 and is given by (( ) ) 4 1 4 = (2 + δ0) 3 + c12 + √ c22 ≤ 222.70, (3.107) log 3 log 3 where ci2 = ci(1/(α4 log 3), 1) are defined in (1.5)(1.6) and with α4 defined in (3.6). Proof. Recall from (3.14) that |H(ρ)| ≤ M(δ,0)|ρ| . As a result ∑ (4) ≤ x β−1 J (x) M(δ, 0) | . (3.108)ρ| ρ∈Z(ζ),ρ=β0,|ρ|≥ 12 |γ|≤ 1 α4 log dL For ρ = β + iγ, (3.6) implies β − 1 ≤ − 1 1α4 log d when |γ| ≤ α log d and ρ = β0. Thus usingL 4 L |ρ| ≥ 1/2, we have ∑ xβ−1 − 1 ∑ ∑≤ ≤ − 1 ∑α log d| | 2x 4 L 1 2x α4 log dL nχ, 1 ,1(0).ρ α4 log dL ρ∈Z(ζ),ρ =β0,|ρ|≥ 1 χ2 ρ∈Z(χ),ρ= β0,|ρ|≥ 1 χ 2 |γ|≤ 1 |γ|≤ 1 α4 log dL α4 log dL (3.109) Similar to (3.105), using Proposition 1.13 with a = 1α log d , = 1, we get4 L ∑ ( ) nχ, 1 ,1(0) ≤ c12 log 3 + c22 nL + c11 log dL + 4c12. (3.110) α4 log dL χ We conclude by combining (3.108), (3.109), (3.110) with (3.67). To provide bounds for J (5)(x, T ) defined in (3.65) and J (6)(x, T ) defined in (3.66), we use 109 ∣∣ ∣∣ 3.6. EXPRESSING ∣∣I − |C|L/K |G|XH(1)∣∣ IN TERMS OF SUM OVER ZEROS the explicit zero free region for ζL(s) as shown in (3.8) with R defined in (3.9). Lemma 3.28. Under the assumptions in (3.67), we have 1 J (5) − (x, T ) ≤  (log d 2 R log(D(T+2)/2)5 L)(log T ) x L , (3.111) where 5 depends on T0 and δ0, and is given by [ ( ) 2 + δ0 log(T0 − 1) 1 2 α1 T0 + 1 5 = 2 ( + + 2α1 + +π(log T )2 (log T )2 π (log T )2(T − 1) (π(T − 1)(log T )20 0 0 0 0 0 1√ 1 (T0 + 1) log(T0 + 1) α1 log(T0 + 1) 1 0.22+ + −)) + −( + )] 2α2 + + 1.3α1log 3 π π(T0 1)(log T )20 (T0 1)(log T )20 (log T0)2 π α2 − 2 log(2πe) 1 α3+ − + 2α3 + ≤ 7.27, (3.112)T0 1 π (log 3)(log T0)2 T0 − 1 with α1, α2, α3 defined in (3.5). Proof. For all ρ = β + iγ with |γ| ≤ T and ρ = β0, by (3.8), we have β − 1 ≤ −(RL(log(D(T + 2)/2))−1. Also, (3.12) and (3.14) with k = 0 gives |H(ρ)| ≤ M(δ,0)|ρ| . Therefore, (5) ≤ − 1 ∑ 1 J (x, T ) M(δ, 0)x RL log(D(T+2)/2) | | . (3.113)ρ ρ∈Z(ζ),|ρ|≥ 12 1 <|γ| T ≥ 2, xRL log(Dτ/2) ≤ xRL log(D(T+2)/2) and x RL log(Dτ/2) ≤ − 1 x RL log(D|γ|) . Finally, we conclude by combining (3.118) and (3.119). 111 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS Proposition 3.30. Under the assumptions in (3.67), we have ∣∣∣∣ − |C|I xH(1)∣∣ ∣ L/K ( |G| ∣ ≤ |C| 1 1 1 xβ0 1− | | H(β ) +  (log d 2 α log d 0 1 L)(log x) + 2(log x)x 2 + 3(log dL) x 2 + 4(log dL)x 4 L G ( )) 2 1− 1 M(δ,m) 1+  (log d )(log T ) x R log(D(T+2)/2) + xR log(D(T+2)/2)S(1)(m,T ) + xS(2)5 L L L (m,T, x) , 2δm where 1, 2, 3, 4 and 5 are defined in (3.92), (3.97), (3.102), (3.107) and (3.112) respectively. Proof. We combine Corollary 3.17 with Lemma 3.24, Lemma 3.25, Lemma 3.26, Lemma 3.27, Lemma 3.28 and Lemma 3.29. Therefore, it remains to estimate S(1) and S(2), which are the sums over the larger zeros. 3.7 Study of the sums over the larger zeros We recall the zero-free region obtained for all Dedekind zeta functions as described in The- orem 3.12: Let ρ = β + iγ be non-trivial zero of ζL(s) with ρ = β0 and τ = |γ|+ 2. Then there exists R > 0 such that 1 β < 1− , (3.120) RL(log(Dτ/2)) 1 n where RL = RnL with R defined in (3.9) and D = 2d LL . Introducing the function − 1 ( − )x RL(logDu) 1 log x φm,x(u) = = exp , (3.121) um+1 um+1 RL(logDu) we rewrite ∑ ∑ S(1)(m,T ) = φ (2)m,1(|γ|) and S (m,T, x) = φm,x(|γ|). (3.122) ρ∈Z(ρ) ρ∈Z(ρ) |γ|≥T |γ|≥T For the rest of the article, we denote ⎨⎪⎧⎪ T if log x ≤ Xm,T , Xm,T = (m+1)RL log 2(DT ), and T1 = ⎪⎪⎩ (√ ) (3.123)W = 1D exp log xR (m+1) if log x > Xm,T .L 112 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS Note that if log x > Xm,T , W > T . We also recall the estimate for the number of zeros NL(T ) given by Theorem 3.10 and define Q(t, u) = P (u) − P (T ) + E(u) + E(T ) so that Q(t, u) is an upper bound for NL(u)−NL(t) : ( ) ( ) ( √ ) nLu Du − nLt Dt D utQ(t, u) = log log + 2α1nL log + 2α2nL + 2α3, (3.124) π 4πe π 4πe 2 with αi’s defined in (3.5). The next lemma provides a bound for Q(t, u) similar to [4, Lemma 2.15] for Dirichlet L-functions. Lemma 3.31. Let Q(t, u) be defined as in (3.124). If 44 ≤ t ≤ u, then unL Q(t, u) < log(Du). π ( ) Proof. Let (t, u,D) = π unLn π log(Du) − Q(t, u) . We have ∂∂u = log(4πe) − α1π and ∂L u ∂D = t−2α1π D , which are both positive as long as u > α1π/ log(2πe) and t > 2α1π. Thus, for u ≥ t ≥ 44, (t, u,D) increases with both u and D, and (t, u,D) ≥ (t, t, 1) with (t, t, 1) = (t−2α1π) log t+ 2α1π log 2− 2α π − 2α3π2 n > 0.L Remark 3.32. Here, the condition t ≥ 44 is implied by (t, t, 1) > 0 and thus depends on the values of the αi’s. 3.7.1 Estimating the sum over inverse of zeros S(1)(m,T ) Lemma 3.33. Under the assumptions in (3.67), we have, log T S(1)(m,T ) ≤ 6(m)(log dL) , Tm where 6(m) depends on m and T0, and is given by ( ) ( )2α1 + 2 α1log 3 m+1 + 2α2 + 2α31 2 1 4α1 6(m) = + + + ≤ 1.38, (3.125) mπ log 3 log T0 (log 3)T0 (log T0)T0 where α1, α2 and α3 are as defined in (3.5). 113 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS Proof. By partial summation, ∑ ∫ ∞ ∫ (1) 1 d(NL(u)−N ∞L(T )) NL(u)−NL(T )S (m,T ) = | | = du = (m+ 1) du.γ m+1 um+1 um+2|γ|>T T T Together with Lemma 3.31, we get ( ( )∫ n D ∞ ∫ du n ∞ ∫ (1) ≤ L L log u ∞ log u S (m,T ) (m+ 1) log + du+ α n du π 4πe um+1 π um+1 1 L m+2 ( T T ( ))∫ T u ) − T − nLT T ∞ du + 2α1 log dL + α1nL log T + 2α2nL + 2α3 log dL log . π π 2πe T u m+2 We calculate the integrals ∫ ∞ ∫1 1 ∞ log u log T 1 du = , du = + , uk+1 kT k uk+1 kT k k2 kT T T and finally obtain ( ( ( ))) nL 1 log T log dL nL (m+ 1) 1 S(1)(m,T ) ≤ + ( + − log 2πeπ m Tm mπ mπ m ) Tm log T α1nL 1 + 2α1nL + + 2α1 log dL + 2α2nL + 2α3 . Tm+1 (m+ 1) Tm+1 We conclude with the bounds on nL, dL from (3.67) and the fact that m+1 − log(2πe) m2π mπ ≤ 0. (2) 3.7.2 Estimating SL (m,T, x) Recall that ∑ − 1x RL log(D|γ|) S(2)(m,T, x) = | | |γ| . m+1 γ >T Lemma 3.34. Under the assumptions (3.67), we have ∫ ∞( ) S(2)(m,T, x) ≤ ∂Q(T, T1)φm,x(T1) + Q(t, u) φm,x(u)du, T ∂u where φm,x, Q, and T1 are defined as in (3.121), (3.124), and (3.123) respectively. 114 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS Proof. Partial summation, NL(u)−NL(T )  u log u and φm,x(u)  u−2 as u → ∞, give ∫ ∞ ∫ ∞ S(2)(m,T, x) = φm,x(u) d(NL(u)−NL(T )) = (NL(u)−NL(T ))(−φ′m,x(u))du. (3.126) T T Note that −φ′m,x(u) ≤ 0 for u ≤ W and that −φ′m,x(u) > 0 for u > W . Using this and NL(u)−NL(T ) ≤ Q(t, u) in (3.126), we get ∫ ∞ S(2)(m,T, x) < Q(t, u)(−φ′m,x(u))du. T1 Again integrating by parts yields ∫ ∞( ) S(2) ∂ (m,T, x) ≤ Q(T, T1)φm,x(T1) + Q(t, u) φm,x(u)du. T ∂u1 Since the last integrand is positive, we can conclude by pushing T1 to T . Lemma 3.35. Under the assumptions (3.67), we have ⎪⎧⎪⎨⎪ ( )2E(T )φ 2E(T ) − log x√m,x (T ) = m+(1 exp R if 0 < log x ≤ XT m,T ,≤ ⎪ L(logDT )Q(T, T1)φm,x(T1) ⎪⎪ )⎩ √n mLD log xπ R (m+1) exp − (2m+ 1) log xL RL(m+1) if log x > Xm,T , where T1 and Xm,T are defined in (3.123). Proof. In the case 0 < log x ≤ Xm,T , T1 = T and thus Q(T, T ) = 2E(T ), giving the first inequality: Q(T, T1)φm,x(T1) = 2E(T )φm,x(T ). In the other case, log x > Xm,T , then T1 = W > T ≥ 44, and Lemma 3.31 gives √ (√ ) WnL nL log x log x Q(T, T1) = Q(T,W ) < log(DW ) = exp . (3.127) π πD RL(m+ 1) RL(m+ 1) Similarly, ( ) ( √ ) 1 log x (m+ 1) log x φm,x(T1) = φm,x(W ) = exp − = Dm+1 exp − 2 . Wm+1 RL(logDW ) RL (3.128) 115 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS √ We conclude by putting together (3.127), (3.128) and 2 m+ 1− √ 1 = √2m+1 . m+1 m+1 To complete the bound for S(2)(m,T, x), it remains to estimate the integral ∫ ∞( )∂ Q(t, u) φm,x(u)du. T ∂u To do so, we introduce the following integral functions. Given positive real numbers n,m, α, β and l, we define an incomplete modified Bessel function of the first kind as ∫ ∞ ( )(log βu)n−1 α In,m(α, β; l) = exp − du. (3.129) um+1l log βu Moreover, given positive constants n, z, and y, we call the “imposter” Bessel function of the second kind the integral ∫ ∞ ( ( ))1 z 1 K (z; y) = vn−1n exp − v + dv. (3.130) 2 y 2 v √ Both integrals are related through the change of variable v = (log(βu)) mα ( )n/2 ( √ ) I (α, β; l) = 2βm α √ m n,m Kn 2 αm, log(βl) . (3.131) m α In particular, in our context, ( ) ( )n/2 ( )log x log x In,m , D;T = 2D m Kn zm, wm , (3.132) RL mRL where √ √ m log x mRL zm = 2 , wm = log(DT ). (3.133) RL log x Lemma 3.36. Under the assumptions (3.67), we have ∫ ∞( )∂ 2nL log x Q(t, u) φm,x(u)du ≤ Dm K2(zm, wm), T ∂u π mRL where K2 and (zm, wm) are defined in (3.130) and (3.133) respectively. 116 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS Proof. It follows from (3.124) that ( ) ∂ nL Du α1nL ≤ nLQ(t, u) = log + log(Du), ∂u π 4π u π since log(4π) − α1π T ≥ 0 for u ≥ T ≥ 44. We recognize ∫ ∞ ( )log x log(Du)φm,x(u)du = I2,m , D, T , T RL and conclude with (3.132). Combining Lemma 3.34, Lemma 3.35 and Lemma 3.36 leads to a bound for S(2) in terms of K2: Lemma 3.37. Under the assumptions (3.67), we have S(2)(m,T, x) ≤ B(2)(m,T, x), where ⎧ ⎪⎪⎪⎪⎪ ⎪ ( ) ⎪ 2E(T ) − log x 2nL m log x⎪⎪ Tm+1 exp RL(logDT ) + π D mR K2(zm, wm) if 0 < log x ≤ Xm,T ,⎪ L⎪⎨ B(2)(m,T, x) = ⎪⎪ ( )⎪⎪⎪ ⎪ √ √⎪ n mLD log x exp − (2m+ 1) log x⎪⎪ π RL(m+1) R⎪ L (m+1) ⎩⎪+2nLDm log xπ mR K2(zm, wm) if log x > Xm,T ,L (3.134) where Xm,T , K2 and (zm, wm) are defined in (3.123), (3.130) and (3.133) respectively. The last section investigates those “Bessel” integrals. 3.7.3 Study of impos√ter Bessel function K2(zm, wm) Lemma 3.38. If w < mm m+1 , then ( ) ( ) ( ) ( ) ≤ 1 1K (z , w ) − w k 1 1 12 m m m zm, + J2a zm, + J2b zm, , (3.135) 2 wm wm wm wm 117 3.7. STUDY OF THE SUMS OVER THE LARGER ZEROS where ⎧ ( ) ⎪⎪⎨⎪ ( ( )) √1 1+ z2 +1w exp − zm 12 w + w 1 mm if 1 ≤ w < z , k 1 m m m m zm, = wm ⎪⎪⎩⎪ √ ( )) √ (3.136) 1+ z2m+1 √ 1+ z2 +1 z exp − z2m + 1 if 1w ≥ m ,m m zm (35y3/2 + 128y + 135y1/2 + 128y−1)z + 105y1/2 + 256 J2a(z; y) = (√ ( , (3.137)256z2e)z)(y+1/y)/2√ z √ − √1 128z2 +√240z + 105J2b(z; y) = πerfc y , (3.138) 2 y 256 2z5/2ez with ∫ 2 ∞ erfc(u) = √ e−t2dt. (3.139) √ π u Proof. We assume w mm < m+1 , i.e. log x > Xm,T . Modifying (3.130) for n = 2, we have ∫ 1 1/w ( ( )) ( ) m zm 1 1 K2(zm, wm) = u exp − u+ du+K2 zm, . (3.140) 2 w 2 u wm m Kadiri and Lumley in [14, Lemma A.4] proved that ∫ 1/w ( ( )) ( ) ( )1 m − zm 1 ≤ 1 1 1u exp u+ du − w m k zm, , (3.141) 2 w 2 u 2 wm m wm where k is defined in (3.142). Also, since 1w > 1, [4, Proposition 4.7] gives usm ( ) ( ) ( ) 1 1 1 K2 zm, ≤ J2a zm, + J2b zm, , (3.142) wm wm wm where J2a and J2b are defined in (3.137) and (3.138) respectively. We conclude by putting together (3.140), (3.141) and (3.142). √ Remark 3.39. The other case, w ≥ mm m+1 , which is the same as log x ≤ Xm,T , will be discussed in following articles. 118 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T Lemma 3.40. If log x > Xm,T , then √ ( √ ) Dm(2) ≤ nL log x − log xB (m,T, x) exp (2m+ 1) π R(m(+(1) ) ( RL)(m+ 1() ) ( )) 2 log x 1 1 1 1 1 + Dm − wm k zm, + J2a zm, + J2b zm, . π mR 2 wm wm wm wm √ Proof. Assuming log x > Xm,T is the same as having w < m m m+1 . We use (3.134) and Lemma 3.38 to complete the proof. 3.8 Explicit formula for the error term in the case log x > Xm,T Let us introduce aβ0 as the following: ⎪⎪⎨⎧ 1 if β0 exists, aβ0 = ⎪⎩⎪ (3.143)2 otherwise. Theorem 3.41. Let C be a conjugacy class of G = Gal(L/K). Let β0 be the possible exceptional real zero of ζL(s). Under the assumptions in (3.67), we have ∣∣∣ − |C| ∣∣ψC(x) | |x ∣∣∣ xβ0−1GEψ(x) = | | ≤ + ψ(δ,m, T, x),C |G|x β0 where δ 2 − 1 M(δ,m) (δ,m, T, x) = +  (δ,m, T, x)(log T ) (log d )x RL log(D(T+2)/2)ψ 7 L + B(2)(m,T, x), aβ0 2δ m (3.144) 0 + 1 −1+ 1  − 1+ 1  (δ,m, T, x) = (log x)x RL log(D(T+2)/2) 2 + (log x)x 2 RL log(D(T+2)/2)7 (log T )2 (log T )2(log dL) 3 − 1+ 1 4 − 1 + 1 + (log d )x 2 RL log(D(T+2)/2) + x α4 log dL RL log(D(T+2)/2)L (log T )2 (log T )2 M(δ,m) 1 −1+ 2 + 5 +  (m) x RL log(D(T+2)/2)6 (3.145) 2δm (log T )Tm and where H, M and B(2) are defined in (3.11), (3.12) and (3.134) respectively, and where the li’s are defined in (3.29)(3.92)(3.97)(3.102)(3.107)(3.112)(3.125). 119 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T Proof. We recall that IL/K = ĨL/K(x) + ψ̃C(x) as defined in (3.25). Therefore E (x) = ∣∣∣ ψ̃ (x)− |C| ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ C |G|x ∣∣ |C| |C| ∣ ∣∣∣IL/K − |G|x≤ ∣∣∣ ∣∣∣ ĨL/K(x) ∣∣ ∣∣IL/K − |G|xH(1) ∣∣ ∣∣ ĨL/K(x) ∣∣ψ̃ |C| | | + ∣ ≤ ∣ ∣+|H(1)−1|+∣x C x |C|x |C|x |C| ∣.|G| |G| |G| |G| |G|x (3.146) β −1 Using (3.14) with k = 0 and x 0β ≤ 1, we obtain0 β0−1 xβ0−1 x δ H(β0) ≤ + . (3.147) β0 2 Combining (3.146), (3.147), |H(1)−1| = δ2 and Lemma 3.13 with Proposition 3.30, Lemma 3.33 and Lemma 3.37, we obtain ≤ x β0−1 Eψ̃(x) + ψ̃(δ,m, T, x),β0 with δ 1 1 ψ̃(δ,m, T, x) = + (0 + 1)(log dL)(log x)x −1 + 2(log x)x− 2 + 3(log d )2x−L 2 aβ0 − 1 1 +  (log d )x α(log d +  (log d )(log T )2 − R log(D(T+2)/2) 4 L 4 L 5 L x L ) M(δ,m) (log dL)(log T ) −1+ 1 +  (m) x RL log(D(T+2)/2) (2)6 +B (m,T, x) . 2δm Tm Note that ψ̃ is independent of α. Hence Eψ(x) ≤ max{E−(x), E+(x)} ≤ Eψ̃(x) completes the proof. Remark 3.42. Assume (3.67) with log dL ≤ log x√ which satisfies the condition for The-4mR(log 3) orem 1.15 given in (3.165). Then 7(δ,m, T, x) ≤ 7(δ,m, T0, x ) ≤ 7.27 for δ = 10−190 , m = 2, T0 = 44, x0 = exp(4484), α4 = 2 and M = MR as defined in (3.18). Going forward, we are going to use 6 and 7 to denote the functions 6(m) and 7(δ,m, T, x) respectively. 120 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T 3.8.1 Explicit bounds for Eψ(x) independent of T and δ but dependent on dL We choose T as the following: ( √ ) 1 T = exp √1 log x . (3.148) D 2 m RL Thus √ ( ) mRL 1 1 1 3 wm = log(DT ) = and − wm = . (3.149) log x 2 2 wm 4 We determine which case this corresponds to in order to define k. We have √ 1 ≥ 1 + z 2 m + 1 ⇐⇒ (2zm − 41)2 ≥ z2 + 1 ⇐⇒ 3z2m m − 4zm ≥ 0 ⇐⇒ zm ≥ .wm zm 3 √ Remember that we are in the region log x > Xm,T , which by using D ≥ 2 3, m ≥ 2 and T ≥ T0 ≥ 44 gives log x √ √ > (m+ 1)(log(DT ))2 ≥ (m+ 1)(log(2 3T ))2 ≥ 3(log 88 3)20 (3.150) RL √ √ and ensures that zm > 2 6(log 88 3) > 4 3 . Hence we are in the case where by (3.142), ( ) √ ( ) √4m log x ( √ )  1 1 + z 2 √ 1 + m + 1 R + 1 L m log x k zm, = exp − z2 + 1 ≤ √ exp − 2 . wm z m m 2 m log x RLRL √ 1+ 4m log x+1 √ √ RUsing (3.150) and L ≤ √ 1 √ + 1 + 1 √ ≤ 1.042, it 2 m log x 4m(m+1)(log(2 3T0))2 4m(m+1)(log(2 3T0)) 2 RL follows ( ) ( ) ( √ ) 2 m log x 1 1D − w  1m k zm, ≤ m log x − m log xηm,1D exp 2 , (3.151) π mR 2 wm wm R RL where for m ≥ 2 and T0 ≥ 44, ( √ ) 3 1 1 ηm,1 = √ √ + 1 + √ ≤ 0.249. (3.152)2πm 2 4m(m+ 1)(log(2 3T ))2 4m(m+ 1)(log(2 3T0))0 121 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T Now using the definition of J2a as in (3.137), we find ( √ ) ( √ ) √ √m log x m log x 205 2 + 320 2 R + 105 2 + 256L J2a 2 , 2 = ( ) R √L (256)(4)m log x 5 m log x [ R exp L 2 RL √ √ √ ]√ ( √ ) 205 2 + 320 105 2 + 256 RL RL 5 m log x = + exp − . (256)(2) (256)(4) m log x m log x 2 RL Using (3.150), we obtain ( √ ) √ ( √ ) 2 m log x m log x ≤ m nL log x 5 m log xD J2a 2 , 2 ηm,2D exp − , (3.153) π mR RL R 2 RL where [ √ √ ] 2 205 2 + 320 105 2 + 256 1 ηm,2 = 3 + √ √ ≤ 0.275 . . . . (3.154) πm 2 (256)(2) (256)(4) m(m+ 1) log(2 3T0) √ (√ ) √z Using the definition (3.138) for J2b, together with zm 2 2 − √1 = m2 and the bound √ 2 πerfc(x) < exp(−x2)x−1(see [1, 7.1.13]), we get 2 128z2 + 240z + 105 2 − 128 + 240z−1m m 5 z m + 105z−2m 2 − 5J (z m zm2b m, 2) < √ √ √ e 4 < √ e 4 . z 5/2m 256 2z zm 256 2 zm m √ √ √ For z = 2 m log xm R > 2 m(m+ 1) log(2 3T0), we haveL √ ( √ ) 2 m log x nL log x 5 m log xD J2b(zm, 2) ≤ η mm,3D exp − , (3.155) π mR R 2 RL with ( 1 240 ηm,3 = √ 128+ √ √ + √ ) 105 √ ≤ 0.086. 128 2πm3/2 (2 m(m+ 1)(log(2 3T0)) (2 m(m+ 1)(log(2 3T0)))2 (3.156) 122 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T Inserting (3.151), (3.153) and (3.155) in Lemma 3.40, we obtain √ ( √ ) ( √ ) m B(2) D nL log x 2m+ 1 log x log x m log x (m,T, x) ≤ exp − m1 + ηm,1D exp − 2π R(m+ 1)√ (m+ 1() 2 √RL ) R RL nL log x 5 m log x + (ηm,2 + ηm,3)D m exp − . R 2 RL √ √ √ As 5 m ≥ √2m+12 ≥ 2 m, and log xR > (m+ 1)(logDT )2 ≥ (m+ 1)(log(2 3T0))2, thenm+1 L ( √ ) B(2) √ log x (m,T, x) < κ Dmm (log x) exp − 2 m , (3.157) RL ( ( 1 √ ) √ ) with κm = √ exp −( 2m+ 1− 2 m(m+ 1) (log(2 3T0))πR(m+ 1)(log(2 3T0)) ) η √m,1 ηm,2 + ηm,√3 1 √+ + √ exp − m(m+ 1)(log(2 3T0)) ≤ 0.009. (3.158) R R m+ 1(log(2 3T0)) 2 Proof of Theorem 1.14. We apply Theorem 3.41 with the choice (3.148) for T ≥ T0 ≥ 44 √ and D ≥ 2 3. More precisely we use √ log x ≥ 4mR (logDT )2L 0 ≥ 4mRL(log 88 3)2 ≥ 11954, (3.159) √ √ − log x − log x 1 log x −2 m √log x (log T )2e RL log(D(T+2)/2) ≤ (log(DT ))2e RL log(DT ) ≤ e RL , 4m RL so that we get for ψ as defined in (3.144): √ √ ≤ δ 1 log x − √ 2 m log x M(δ,m) − √2 m log x ψ(δ,m, T, x) +  (log d ) e Rn m 7 L L + κmD (log x)e RnL . aβ0 4m RnL 2δ m (3.160) Denoting √ M(δ,m) √m −2 m log xA = κmD (log x)e RnL , (3.161) 2 we choose δ such that δ + Aδ−ma with is as small as possible. We check that δ −m β a + Aδ 0 β0 123 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T 1 minimizes at δ = (aβ0mA)m+1 . With this choice of δ, we obtain ( ) δ − 1 − m − m 1+Aδ m = mm+1 +m m+1 a m+1Am+1 aβ0 ( β0 ) 1 √ √ − mM(δ,m)κ Dm m+1 1 −2 m log xm= (1 +m 1) (log x)m+1 e m+1 RnLm ,2aβ0 and thus ( ) 1 mM(δ,m)κ Dmm m+1 ψ(δ,m, T, x) ≤ (1 +m−1) 2amβ0 √ √ ) √ √ 7 log dL 1− 1 −2m m log x 1 −2 m log x+ (log x) m+1 e m+1 RnL (log x)m+1 e m+1 RnL . (3.162) 4mR nL 3 √ 1 − 2m 2 log x Since log x ≥ 4mRnL(logDT )2 ≥ RnL0 m , then (log x)1−m+1 e m+1 RnL decreases with x. Hence 3 √ 1− 1 − 2m 2 log x ( √ ) 1 √ 4m2(log x) m+1 e m+1 Rn 2L ≤ (4mRnL log(2 3T0) )1−m+1 (2 3T0)−m+1 , 1 √ n as D = 2d LL ≥ 2 3. We deduce ( ( ) 1 mM(δ,m)κ Dmm m+1 ψ(δ,m, T, x) ≤ (1 +m−1) ( 2a m √ )β02− 2 √ 4m2 ) √ √ log(2 3T ) m+1 (2 3T )−0 0 m+1 1 −2 m log x + 7(log dL) (log x)m+1 e m+1 RnL1 ( (4mRnL)m+1 √ √1≤ m − ) 1 −2 m log xmax Dm+1 , (log d )n m+1 λ(m)(log x)m+1 e m+1 RnL LL with ( ) ( √ ) − 2 √ − 4m21 2 m+1 m+1 m+1 λ(m) = (1 +m−1 mM(δ0,m)κm log(2 3T0) (2 3T0) ) + 7 , (3.163) 2am 1β0 (4mR)m+1 where λ(m) ≤ 0.787 if β0 exists and λ(m) ≤ 0.496 otherwise. Proof of Corollary 3.2. We set R = 29.57, T = 44 and M (δ,m) = (m((1 + δ /m)m+10 R 0 + √ 1))m. Using GP/Pari, we optimize m and δ0 to obtain m = 2, δ = 10 −23 0 , 2 m m+1 ≥ 0.942 and the √ explicit constant λ(2) = 0.782 if β0 exists and obtain m = 2, δ = 10 −19, 2 m0 m+1 ≥ 0.942 and the 124 3.8. EXPLICIT FORMULA FOR THE ERROR TERM IN THE CASE LOGX > XM,T explicit constant λ(2) = 0.493 if β0 does not exist. Using this, we obtain the required result. 3.8.2 Explicit bounds for Eψ(x) independent of T , δ and dL Proof of Theorem 1.15. Using our choice of T as in (3.148) and T ≥ T0, we obtain: ( √ ) 1 n ≤ 1 1 log xD = 2d LL exp , (3.164)T 10 2m 2 RnL and √ 1 nL log x log dL ≤ 1 . (3.165) 2m 2 R Thus, (3.162) is modified into ( ( √ √ ) 1 ≤ −1 mM(δ,m)κ 1 m log x m m+1 2 Rnψ(δ,m, T, x) (1 +m ) e L√ 2amβ Tm0 0 √ √ ) √ √  1 n log x 1 −2m m log x 1 −2 m log x7 L + 1 (log x) 1− m+1 e m+1 RnL (log x)m+1 e m+1 RnL 4(mRnL 2m 2( R ) 1 ≤ −1 mM(δ,m)κm m+1 (1 +m ) 2am Tmβ0 0 3 1 √ ) √ √  3 1 − 2m 2 +0.5m 2 log x7 1 −1.5 m log x + 1 (log x) − 2 m+1 e m+1 RnL (log x)m+1 e m+1 RnL . (3.166) 3 8(mR) 2n 2L ( ) 3 1 √2 3 1 − 2m 2 +0.5m 2 log x Since log x ≥ 4mRL log(DT0) ≥ RnL , then (log x) −2 m+1 e m+1 RnLm decreases with x. Hence 3 1 √ 3 1 − 2m 2 +0.5m 2 log x ( √ ) 3 1 √ 4m22 +m (log x) −2 m+1 e m+1 RnL ≤ (4mRL log(2 3T0) ) −2 m+1 (2 3T −0) m+1 , 1 n as D = 2d LL ≥ √ 2 3. We deduce ( ( ) 1 ≤ −1 mM(δ,m)κ 1 m m+1 −1+ ψ(δ,m, T, x) (1 +m ) m+1 m m nL 7 ( ) 2aβ T0 0 √ √ ) √ √23− 2 4m +m 1− 1 1 −1.5 m log x + log(2 3T0) m+1 (2 3T ) − m+1 n m+10 (log x)m+1 m+1 RnL1 L e (4mR)m+1 √ √ ≤ 1− 1 1 −1.5 m log x ν(m)n m+1 m+1 m+1 RnLL (log x) e 125 3.9. EXPLICIT BOUNDS FOR PRIME IDEAL COUNTING FUNCTIONS IN CHEBOTAREV’S DENSITY THEOREM EQUIVALENT TO θ AND π with 1 +m− ( ) 1 1 mM(δ 20,m)κm m+1 7 ( √ )3− 2 √ 4m +m ν(m) = + log(2 3T ) m+1 −0 (2 3T0) m+1 , 2 am m 1β T0 0 (4mR)m+1 (3.167) where ν(m) ≤ 0.0398 if β0 exists and ν(m) = 0.0251 otherwise. Proof of Corollary 1.16. We set R = 29.57, T0 = 44 and M(δ0,m) = MR(δ0,m) = (m((1 + 1 δ/m)m+1 + 1))m. Using GP/Pari, we optimize m and δ to obtain m = 2, δ = 10−19, 1.5m 2m+1 ≥ 0.707 and the explicit constant ν(m) = 0.0396 if β0 exists and obtain m = 2, δ = 10 −23, 1 1.5m 2 m+1 ≥ 0.707 and the explicit constant ν(m) = 0.0249 if β0 does not exist. Using this, we obtain the required result. 3.9 Explicit bounds for prime ideal counting functions in Chebotarev’s den- sity theorem equivalent to θ and π Remember that: ∑ πC(x) = 1. (3.168) p unramified σp=C, N p≤x To go from ψC(x) to πC(x), we introduce new quantities as ∑ ∑ θC(x) = log(N p) and θ0(x) = log(N p). (3.169) p unramified p unramified σp=C, N p≤x N p≤x Lemma 3.43. For x ≥ x0, | 22 √ψC(x)− θC(x)| < nK x(log x). 15 Proof. Since N p ≥ 2, therefore for N pm ≤ x, m ≤  log xlog 2 . Using this, we notice that ∑ ( 1 )1 1 log x log(N p) = θ0(x 2 ) + θ0(x 3 ) + · · ·+ θ0 x  log 2 . (3.170) p unramified m≥2, N pm≤x Clearly, θC(x) ≤ θ0(x). Also, it can be easily verified that θC(x) ≤ θ0(x) ≤ nKθ(x) where θ(x) = 126 3.9. EXPLICIT BOUNDS FOR PRIME IDEAL COUNTING FUNCTIONS IN CHEBOTAREV’S DENSITY THEOREM EQUIVALENT TO θ AND π ∑ p≤x log p. Rosser and Schoenfeld in [24, Theorem 9] showed that for x ≥ 2, θ(x) < 1.01624x. Combining this we obtain, θ0(x) < 1.01624nKx and ∑ ( 1 ) | 1 1 log xψC(x)− θ  log 2C(x)| ≤ log(N p) = θ0(x 2 ) + θ0(x 3 ) + · · ·+ θ0 x p unramified m≥2, N pm≤x ≤ log x 1 log x × 1 22 √θ0(x 2 ) < (1.01624nKx 2 ) < nK x(log x). (3.171) log 2 log 2 15 Lemma 3.44. Let C be a fixed conjugacy class of the Galois group, Gal(L/K) = G. Let β0 be the possible exceptional real zero of ζL(s). Let H be defined in (3.11). For log x ≥ 19 810n (log d 2L) ,L we have ∣∣θ |C| ∣C(x)− | |x ∣ xβ0−1G Eθ(x) = ∣∣ ∣| | ∣ < + θ(x, nL), (3.172)C | |x β0G where ( √ ) 22 lo√g x 1 2 log x θ(x, nL) = nL +A1(log x) 3n 3 exp −B1 . (3.173) 15 x L nL Proof. Using Lemma 3.43, we obtain ∣∣∣ | | ∣∣ − C ∣ ∣ ∣ ∣ ∣ θC(x) | |x ∣∣ ≤ |θC(x)− | ∣ |C| ∣ 22 √ ∣ |C| ∣ψC(x) + ∣∣ψC(x)− ∣ ∣ ∣G | |x∣ < nK x(log x) + ∣ψC(x)− x .G 15 |G| ∣ (3.174) We complete the proof using |C| ≥ 1, nK × |G| = nL and Remark 1.16. Proof of Theorem 1.17. Using partial summation and integration by parts, we obtain | | ∫ ( ∫ )− C θC(x) x θC(t) − |C| x x dtπC(x) | | Li(x) = + dtG log x 2 t(lo∫g t) 2 |G| +log x 2 (log t)2 θC(x)− |C|| xG|x θC(t)− |C||G| t = + dt. (3.175) log x 2 t(log t) 2 Note that the first equality defined in (3.175) is true up to a constant ≤ nK which depends on #{p|p unramified, Np = 2 and σp = C}. Since this quantity is very small, we neglect this in our computations. For log x ≥ 2×19 810n (log dL)2, using the triangle inequality and Lemma 3.44, weL 127 3.9. EXPLICIT BOUNDS FOR PRIME IDEAL COUNTING FUNCTIONS IN CHEBOTAREV’S DENSITY THEOREM EQUIVALENT TO θ AND π obtain: ∣∣∣∣π (x)− |C| Li(x)∣∣ ∣ C ∣ |G| ∣∣∣θ (x)− |C| ∣ ∫ ∣ |C| ∣ ( ) ∫ ∣ |C| ∣≤ ∣ C |G|x ∣∣log x ∣ x ∣θC(t)− |G| t ∣ |C| x xβ0−1 x ∣θC(t)− |G| t ∣ + ∣∣ ∣∣dt ≤ | | + θ(x, n ∣ ∣L) + ∣ ∣dt | | ( 2 t(log t) 2 G log x ( β 20 √ )) ∫ 2∣ t(log t)|C| ∣ C x xβ0−1 22 log x 1 2 log x x ∣θC(t)− |G| t ∣ = | | + nL √ +A1(log x) 3n 3 exp −B1 + ∣ ∣dt G log x β 15 x L0 n ∣ 2 ∣L 2 t(log t) (3.176) ∣∣ ∣∣ We know θ (t) ≤ θ (t) < 1.01624n t. Therefore ∣∣θ (t) − |C|C 0 K C |G| t∣∣ < 2.01624nKt. Using this and nK × |G| √ = nL after splitting the above integral at x, we obtain: ∫ ∣∣∣ − |C| ∣ ∫ √ ∣x θ (t) ∫ |C| ∣∣ C |G| t ∣∣∣ x 2.01624 x ∣θC(t)− |G| t ∣dt < nK dt+2 2 √ ∣∣ ∣dt2 ∣ 2 t(log t) 2 (log t)∫ ∣ x t(log| | √ ∣∣ − |C| ∣ t) C x ∣θC(t) |G| t ∣< 4.2 | |nL x+G √ ∣∣dt. (3.177)x t(log t)2 √ For the integral on the right side, we can use Lemma 3.44 as log x ≥ 19 810 2n (log dL) . Thus, weL obtain: ∫ x ∣∣∣∣θC(t)− |C| |G| t ∣∣∣ | | ∫ β −1∣ C x t 0 ∫ x β0 |C| θ(t, nL) √ dt < | | √ dt+ dt. (3.178)t(log t)2 G (log t)2 |G| √x x x (log t)2 Now, ∫ √x ∫ ( ( )) θ(t, n xL) 22 lo√g t 1 2 − log t 1√ dt = √ nL +A1(log t) 3n 3 exp B1 dt x (log t) 2 x ∫15 t ∫L n( (√log t)2Lx x )22n 2L √ 1 − 5 − log t= √ dt+A n 31 L √ (log t) 3 exp B1 dt15 x t(log t) x ( √ ) n√ L ≤ 88nL x 5 2 5 log x + 2 3A n 31 L(log x) − 3x exp −B1 . (3.179) 15 log x 2nL Also we can easily check that ∫ xβ0 1 x tβ0−1 + dt ≤ Li(xβ0). (3.180) log xβ0 β √ 20 x (log t) 128 3.10. COMPARISON TO RESULTS FROM WINCKLER Now combining (3.176), (3.177), (3.178), (3.179) and (3.180), we obtain ∣∣∣∣ − |C| ∣π (x) Li(x)∣ ∣ C |G| ∣ |C| | | √ ( )C 22 88 < | | Li(x β0) + G | n x( | LG √ + 4.2 + 15 | | )( 15(log x) ( ( )√ )) C 2 3 − √B1 log x 5 − 5 − 2 − − √B1 log x+ | |A1nLx exp 2 3 (log x) (3 + (log x) 3 exp B√ 1G 2 (nL ) ( ) 2 nL|C| β |C| B1 log x 1 22 88= 0| | Li(x ) +G ( √ |G| nLx exp − √ √ + 4.2 + × ) ( 2 nL x 15 ( 1(5(log x) )√ ))) B log x − 1 exp √1 5 − 5 2 B1 log x+A1n 3 2 3L (log x) 3 + (log x)− 3 exp − B1 − √ .2 nL 2 nL (3.181) √ Using log x > 2× 8R(log 88 3)2n , log x ≥ 11954, nL ≥ 2,L ( √ ) ( ) √1 √B1 log x ≤ 1 B √ 1 √ exp √ exp √ ( 16R(log 88 3)) , x 2 nL exp(16R(log 88 3)2)) 2 with R = 29.57, A1 = 0.0249, B1 = 0.13 (β0 does not exist) we obtain ∣∣∣ ∣ ( √ )∣ |C| ∣ |C| 0.13 log xπC(x)− | | Li(x)∣∣ ≤ 2.97× 10−6G |G|nLx exp − √ , (3.182)2 nL and with R = 29.57, A1 = 0.0396, B1 = 0.13 (β0 exists) we obtain ∣∣∣ | ∣∣ C| ∣ | ( √ ) C| π (x)− Li(x)∣∣ ≤ Li(xβ0) + 4.714× 10−6 |C| 0.13 log xC | | | | | |nLx exp − √ . (3.183)G G G 2 nL 3.10 Comparison to results from Winckler √ We use R = 29.57, log dLn ≥ log 3, nL ≥ 2 and m = 2 to obtainL ( )2 ( ) ( ( ) )1 n 1 log d 2 L log 88n 2 L 4mRnL(log 88d L 2 L ) = 8RnL log 88 + log dL = nL 8(29.57) + 1( ( ) ) nL nL log dL ≤ (log d 2 2 L) log 8(29.57) √88 19 810+ 1 < (log d )2L . nL log 3 nL 129 3.10. COMPARISON TO RESULTS FROM WINCKLER 1 n Thus we obtain the form log x ≥ 19 810n (log d )2L > 4mRnL(log 88d L 2L ) as given in Corollary 3.2L and Corollary 1.16. Now to compare our results with Winckler’s [30], we look at the case which takes into account existence of β0 and work with another form for the above expression. We use R = 29.57, log dL ≥ log 3, nL ≥ 2 and m = 2 to obtain ( )2 ( ( )2)1 n 1 4mRn (log 88d L )2 = 8Rn log 88 + log d = n (log d )2 log 88 1 L (L ( L ) ) L L L 8(29.57) + nL log dL nL log 88 1 2≤ nL(log d 2L) 8(29.57) + < 4953nL(log d 2L) . (3.184) log 3 2 1 √ n Thus log x ≥ 4953nL(log dL)2 implies log x > 4mRnL(log 88d LL )2. Thus using log dL ≥ nL log 3, we obtain √ √ log x ≥ 4953nL(nL log 3)2 = 4953(log 3)2n3L. Using this and then computing in MAPLE, for log x ≥ 11954, we get, ( √ ) 1 2 log x 0.0396(log x) 3n 3L exp − (0.130−B2) ( nL √2 √ ) ≤ 1 (log x) 9√ − − log x 10.0396(log x) 3 2 exp (0.130 B2) 1 (4953(log 3)2) 3 (4953(log 3)2) 9 (log x) 3 0.0396 5 √ 1 1 = √ (log x) 292 exp(−(0.130−B2)(4953(log 3) ) 6 (log x) 3 ) (4953(log 3)2) 9 ≤ A2, where (A2, B2) as given in Corollary 3.5 have the admissible values (1.84 × 10−4, 0.014) and (3.11, 0.125). Thus ( √ ) ( √ ) 1 2 log x 1 2 log x 0.0396(log x) 3n 3L√exp − 0.707 ≤ 0.0396(log x) 3n 3 ( ) Rn L exp − 0.13 L nL ≤ − log xA2 exp B2 . nL Winckler in [30] proved that for log x ≥ 1545n 2L(log dL) , xβ0− ( √ ) 1 ≤ log xEψ(x) + 1.51× 1012 exp − 0.014 . β0 nL 130 3.10. COMPARISON TO RESULTS FROM WINCKLER We instead prove that for log x ≥ 4953nL(log dL)2, − ( √ )xβ0 1 log x Eψ(x) ≤ + 1.84× 10−4 exp − 0.014 . β0 nL Using (3.184), we have 1 × n2 4mRnL(log 88d LL )2 < 9 906n 2L(log dL) . (3.185) 1 n Thus log x ≥ 9 906nL(log d )2L implies log x > 8mRnL(log 88d L )2L . Thus using log dL ≥ √ nL log 3, we obtain √ √ log x ≥ 9 906n (n log 3)2 = 9906(log 3)2n3L L L. Using this and then computing in MAPLE, for log x ≥ 2× 11954, we get, ( ( )√ ) 0.13 log x 4.714× 10−6nL exp − √ − F1 2 ( nL1 ( )√ ) ≤ × −6 (log x) 3√ − 0√.13 − log x √ 1 4.714 10 1 exp F1 1 (9 906(log 3) 2) 3 (9 906(log 3)2) 3 2 (log x) 3 ≤ E1, where admissible values of (E1, F1) as given in Corollary 1.18 are (1.23×10−9, 1/99) and (1.65× 10−5, 0.09). Thus | | ( √ ) | | ( √ )× −6 C − 0√.13 log x ≤ C − log x4.714 10 | |nLx exp E1 | |x exp F1 .G 2 nL G nL Winckler in [30] proved that for log x ≥ 3 090n (log d )2L L , ∣∣∣ ∣ ( √ )∣ |C| ∣πC(x)− ∣| | Li(x)∣ ≤ |C| 1 log xLi(xβ0| | ) + 7.84× 1014x exp − .G G 99 nL We instead prove that for log x ≥ 9 906nL(log dL)2, ∣∣∣ | | ∣∣∣ | | | | ( √ )∣ C C C 1 log xπC(x)− β0 −9| | Li(x)∣ ≤ | | Li(x ) + 1.23× 10 | |x exp − .G G G 99 nL 131 Chapter 4 Future work This chapter provides a brief introduction into plans for future works based on this thesis. 1. Recall that in this thesis, I use the zero-free region for the Dedekind ζ-function as proved by Ahn and Kwon in [3, Proposition 6.1]: Theorem 3.12. Let L be a number field with nL ≥ 2. Let ρ = β + iγ be non-trivial zero of ζL(s) with ρ = β0 and τ = |γ|+ 2. Then β < 1− (RnL log(Dτ/2))−1, (4.1) 1 n where R = 29.57, D = 2d LL . Recently, Lee in [16, Theorem 1] has improved the zero-free region for the Dedekind ζ- function, ζL(s) by proving that Theorem 3.12 holds true for R = 12.2411. I plan to incorporate this new result into my current research to improve the bounds for the error term Eψ(x) as well as increasing the range of x for which the bounds are valid. 2. In this thesis, I have proved unconditional bounds for Eψ(x). However, stronger bounds are known under the assumption of Artin’s Holomorphy Conjecture. This conjecture asserts that: Let L/K be a normal extension with Galois group G. If ρ is a non-trivial irreducible representation of G, then the Artin L-function L(s, ρ, L/K) is a holomorphic function. 132 4. FUTURE WORK I plan to use ideas from Ng [18, Chapter 3] to provide conditional bounds for Eψ(x) assuming Artin’s Holomorphy Conjecture. 3. The study of distribution of zeros of L-functions is critical to finding explicit estimates related to the distribution of prime ideals in number fields. 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