The correction factor in Artin type problems

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Fakhari, Milad
University of Lethbridge. Faculty of Arts and Science
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Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
In 1927, Emil Artin conjectured a product expression for the density of primes p for which a given non-zero integer a is a primitive root modulo p. The conjectured density was proved in 1967 by Hooley under the assumption of the Generalized Riemann Hypothesis. In 2014, Lenstra, Moree, and Stevenhagen introduced a method involving character sums to deduce the formula for the product in the density for Artin’s conjecture. The method applies in similar problems such as the density of primes of cyclic reduction for Serre curves. In this thesis, we introduce a generalization of this method which yields product expressions for a large family of problems that can be stated by summations involving the orders of certain finite groups. As a consequence, the product expressions of some Artin type problems, such as the Titchmarsh Divisor Problem in Kummer families for primes in a given arithmetic progression, are computed here.
number theory , character sums method , density theorems , primes in congruence classes , algebraic number theory , Number theory , Algebraic number theory , Numbers, Prime , Dissertations, Academic