Notes on low discriminants and the generalized Newman conjecture

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 ΛKr (generalizing the de Bruijn-Newman constant Λ) such that for time t≥ΛKr all such L-functions have all their zeros on the critical line; for time t<ΛKr there exist zeros off the line. Under GRH, ΛKr≤ 0; we make the complementary conjecture 0≤ ΛKr. 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<ΛKr. 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≤ ΛKr.