Quadratic residues and the distribution of prime numbers

Abstract

Abstract. D. SHANKS [11 ] has given a heuristical argument for the fact that there are "more" primes in the non-quadratic residue classes mod q than in the quadratic ones. In this paper we confirm SHANKS' conjecture in all cases q < 25 in the following sense. If l, is a quadratic residue, 12 a non-residue modq, e (n, q, l,, ~z) takes the values -F 1 or -l according to n = I I or l~ mod q, thenfor 0 ~ ~ < 1/2. In the general ease the same holds, if all zeros ~ = fl + i y of all L (s, :~ mod q), q fix, satisfy the inequality fl" -7 "~ < l/4.

Small zeros of Dirichlet L-functions of quadratic characters of prime modulus

“…where Li(X) is defined by (2). We compute these diagonal terms and complete the sums by extending the ranges of the summation variables to infinity.…”

“…In another direction, if L(s, χ) is the Dirichlet Lfunction associated to the non-principal character χ modulo 4 then the first low zeros of L(s, χ) dictate how the primes are distributed in residue classes 1 and 3 (mod 4). For further details on how low zeros of Dirichlet L-functions are related to problems in number theory we refer the reader to [2,3,16,21].…”

Small zeros of quadratic L-functions

“…It is well known (see [3,4,8,11]) that the zeros of Dirichlet .L-functions L(s, x) close to the real axis contain significant number-theoretic information. For example if X is a quadratic character with x(~l) = ~1 | then zeros of ^(a;, x) close to s = 1/2 have an effect on the class numbers of complex quadratic fields.…”

“…Obviously the sense in which this predominance occurs needs to be specified. Bentz [4] and Bentz and Pintz [3] have made progress in 308 A.E. Ozliik and C. Snyder [2] this direction.…”

Comparative prime number theory: A survey

“…and when dealing with quadratic residues and distribution of primes, H. Bentz [41] assumes the following two conjectures: Conjecture 5.11 (R 2 ). The domain σ > 1 2 , |t| ≤ 1 is zero free and there is NO zero at s = 1 2 for Dirichlet L-function.…”