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 $\Lambda_{Kr}$ (generalizing the de Bruijn-Newman constant $\Lambda$) such that for time $t\ge\Lambda_{Kr}$ all such $L$-functions have all their zeros on the critical line; for time $t<\Lambda_{Kr}$ there exist zeros off the line. Under GRH, $\Lambda_{Kr}\le 0$; we make the complementary conjecture $0\le \Lambda_{Kr}$. Following the work of Csordas \emph{et. al}. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet $L$-functions to show that $-1.13\cdot 10^{-7}<\Lambda_{Kr}$. 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 $0\le \Lambda_{Kr}$.