Notes on Low discriminants and the generalized Newman conjecture

Abstract

Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet L-functions. There is a real constant Λ_Kr (generalizing the de Bruijn-Newman constant Λ) such that for time t>=Λ_Kr all such L-functions have all their zeros on the critical line; for time t<Λ_Kr there exist zeros off the line. Under GRH, Λ_Kr<=0; we make the complementary conjecture 0<=Λ_Kr. Following the work of Csordas et. al. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet L-functions to show that -1.13* 10^{-7}<Λ_Kr. In the last section we develop a precise definition of a Low discriminant which is motivated by considerations of random matrix theory. The existence of infinitely many Low discriminants would imply 0<=Λ_Kr.