An explicit version of Chebotarev's density theorem

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Das, Sourabhashis
University of Lethbridge. Faculty of Arts and Science
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Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
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 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.
Algebraic number theory , Analytic number theory, mathematical analysis and their applications , Chebotarev, N. G. (Nikolaĭ Grigorʹevich), 1894-1947 , Dissertations, Academic , Lejeune Dirichlet, Peter Gustav, 1805-1859 , Numbers, Prime