## Functiones et Approximatio Commentarii Mathematici

### Notes on low discriminants and the generalized Newman conjecture

Jeffrey Stopple

#### 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 $0\le \Lambda_{Kr}$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 23-41.

Dates
First available in Project Euclid: 24 September 2014

https://projecteuclid.org/euclid.facm/1411564614

Digital Object Identifier
doi:10.7169/facm/2014.51.1.2

Mathematical Reviews number (MathSciNet)
MR3263068

Zentralblatt MATH identifier
1357.11078

#### Citation

Stopple, Jeffrey. Notes on low discriminants and the generalized Newman conjecture. Funct. Approx. Comment. Math. 51 (2014), no. 1, 23--41. doi:10.7169/facm/2014.51.1.2. https://projecteuclid.org/euclid.facm/1411564614

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