Generalised root identities for zeta functions of curves over finite fields

TL;DR

This paper investigates generalized root identities for zeta functions of curves over finite fields, confirming their validity numerically and analyzing their implications for the Riemann hypothesis, highlighting differences from the classical Riemann zeta function.

Contribution

It demonstrates that root identities hold for , -1, and -2 for , and shows that the  root identity aligns with Weil's proof of the Riemann hypothesis for these functions.

Findings

Root identities verified numerically for , -1, -2.

The  root identity is consistent with Weil's proof of RH.

The structure of the counting function N(T) influences the root identities and their implications.

Abstract

We consider generalised root identities for zeta functions of curves over finite fields, \zeta_{k}, and compare with the corresponding analysis for the Riemann zeta function. We verify numerically that, as for \zeta, the \zeta_{k} do satisfy the generalised root identities and we investigate these in detail for the special cases of \mu=0,-1\:\&\:-2. Unlike for \zeta, however, we show that in the setting of zeta functions of curves over finite fields the \mu=-2 root identity is consistent with the Riemann hypothesis (RH) proved by Weil. Comparison of this analysis with the corresponding calculations for \zeta illuminates the fact that, even though both \zeta and \zeta_{k} have both Euler and Hadamard product representations, it is the detailed structure of the counting function, N(T), which drives the Cesaro computations on the root side of these identities and thereby determines the implications of the root identities for RH in each setting.