An extension of the Linnik phenomenon

Abstract

Abstract:We shall prove an extension of Yu.V. Linnik's phenomenon concerning C.L. Siegel's exceptional zeros of Dirichlet L-functions. Also we shall prove a new version of the Brun-Titchmarsh theorem. These two subjects are closely related to each other. The basic tool is our theory of A. Selberg's Λ 2 -sieve, which is developed in our old Tata lecture notes [12]. Constants which are involved either explicitly or implicitly in our discussion are all effectively computable, save for the one in the statement of a theorem of Siegel which has no relation with our main assertions. We add that the discussion in Sections 1-4 is a substantially revised and augmented version of Section 9.3 of our recent monograph [15, Vol. I] and that of Section 5 an extraction from our article [16] posted in arXiv.Keywords: Siegel's zeros; Λ 2 -sieve; Brun-Titchmarsh theorem 1. Our motivation in a historical perspective. We shall first make explicit our notion of exceptional zeros: Let χ denote a genetic Dirichlet character, with which the L-function L(s, χ) is associated. Let T > 0 be sufficiently large, and let Z T = {ρ = β + iγ} be the set of all non-trivial zeros ρ in the region |Im s| ≤ T of the function q≤T * If β T exists, it should be real and simple, and we designate both itself and the relevant unique primitive character χ T , with L(β T , χ T ) = 0, as T -exceptional. It is known that χ T is real. As a matter of fact, we have more preciselyThe upper bound is a consequence of the Dirichlet class number formula for quadratic number fields. However, the formula can be dispensed with. See [7, Section 2, Chapter IX] as well as [15, Section 4.5, Vol. I]; in both monographs are comprehensible accounts of the theory of the zeta and Dirichlet L-functions and the theory of the distribution of primes. One of the most tantalising problems in number theory is the elimination of the possibility of the existence of exceptional zeros. It is generally believed that they do not exist at all. A way to confirm this is to improve the Brun-Titchmarsh theorem in the manner to be made explicit at a later part of Section 5, which remains, however, to be one of the most difficult problems in number theory. Also one may have a hope in employing our Theorem 1 for this purpose; we shall try to be precise in one of the remarks there.