Truncated Product Representations for $L$-Functions in the Hyperelliptic Ensemble

TL;DR

This paper develops truncated product representations for quadratic Dirichlet L-functions over function fields, demonstrating their effectiveness in approximating the functions and analyzing their zeros, including a proof of the Riemann hypothesis for related approximations.

Contribution

It introduces hybrid formulas and explicit argument expressions for these L-functions, and constructs approximations that satisfy the Riemann hypothesis with zeros closely matching the original functions.

Findings

Partial Euler products approximate L-functions away from zeros.

Zeros of approximations cluster near those of the original L-functions.

Almost all zeros of the approximations are simple when the parameter is small.

Abstract

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros, and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions, prove the Riemann hypothesis holds for them, and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.