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

Abstract

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function $\hat{I}\P$, and proved there is a real constant $\hat{I}›$ which encodes the movement of the nontrivial zeros of $\hat{I}\P$ under the deformation. The Riemann hypothesis is equivalent to the assertion that $\hat{I}›\le 0$. Newman, however, conjectured that $\hat{I}›\ge 0$, 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: ${\sup }_{f\in \mathcal{F}}\hat{I}{›}_f\ge 0$, 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 $\hat{I}{›}_D=0$, and thereby prove a stronger statement: ${\max }_{L\in \mathcal{F}}\hat{I}{›}_L=0$. Using geometric techniques, we show a certain deformed $L$-function must have a double root, which implies $\hat{I}›=0$. 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 $\hat{I}›=0$.

Version 1.1, 12 pages