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>=\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<=0; we make the complementary conjecture 0<=\Lambda_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}<\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<=\Lambda_Kr.