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September 2014 Notes on low discriminants and the generalized Newman conjecture
Jeffrey Stopple
Funct. Approx. Comment. Math. 51(1): 23-41 (September 2014). DOI: 10.7169/facm/2014.51.1.2


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 $0\le \Lambda_{Kr}$.


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Jeffrey Stopple. "Notes on low discriminants and the generalized Newman conjecture." Funct. Approx. Comment. Math. 51 (1) 23 - 41, September 2014.


Published: September 2014
First available in Project Euclid: 24 September 2014

zbMATH: 1357.11078
MathSciNet: MR3263068
Digital Object Identifier: 10.7169/facm/2014.51.1.2

Primary: 11M20
Secondary: 11M26, 11M50, 11Y35, 11Y60

Rights: Copyright © 2014 Adam Mickiewicz University


Vol.51 • No. 1 • September 2014
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