Newman's conjecture, zeros of the L-functions, function fields

TL;DR

This paper proves the modified Newman's conjecture for several families of L-functions over function fields, showing that some have the critical parameter exactly zero, which relates to the distribution of their zeros.

Contribution

It extends previous results by confirming the conjecture for specific L-function families and identifies cases where the parameter is exactly zero, strengthening the understanding of zero distributions.

Findings

Proved the modified Newman's conjecture for multiple L-function families.

Identified specific L-functions with the parameter exactly zero.

Used geometric techniques to show certain L-functions have a double root, implying zero parameter.

Abstract

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function $\zeta$, and proved there is a real constant $\Lambda$ which encodes the movement of the nontrivial zeros of $\zeta$ under the deformation. The Riemann hypothesis is equivalent to the assertion that $\Lambda\leq 0$. Newman, however, conjectured that $\Lambda\geq 0$, remarking, "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Andrade, Chang and Miller extended the machinery developed by Newman and Polya to $L$-functions for function fields. In this setting we must consider a modified Newman's conjecture: $\sup_{f\in\mathcal{F}} \Lambda_f \geq 0$, for $\mathcal{F}$ a family of $L$-functions. We extend their results by proving this modified Newman's conjecture for several families of $L$-functions. In contrast with previous work, we are able to exhibit specific $L$-functions for which $\Lambda_D = 0$, and thereby prove a stronger statement: $\max_{L\in\mathcal{F}} \Lambda_L = 0$. Using geometric techniques, we show a certain deformed $L$-function must have a double root, which implies $\Lambda = 0$. For a different family, we construct particular elliptic curves with $p + 1$ points over $\mathbb{F}_p$. By the Weil conjectures, this has either the maximum or minimum possible number of points over $\mathbb{F}_{p^{2n}}$. The fact that $#E(\mathbb{F}_{p^{2n}})$ attains the bound tells us that the associated $L$-function satisfies $\Lambda = 0$.