A refined random matrix model for function field L-functions

TL;DR

This paper introduces a refined random matrix model for a family of function field L-functions, combining properties of random matrices and Euler products to better predict moments and coefficient expectations.

Contribution

It develops a new probability distribution that interpolates between matrix and Euler product models, improving the approximation of L-function statistics.

Findings

The model's low-degree polynomial expectations match Euler product predictions.

High-degree polynomial expectations align with random matrix theory.

Absolute power expectations agree with established conjectures for L-functions.

Abstract

We propose a refinement of the random matrix model for a certain family of $L$-functions over $\mathbb F_q[u]$, using techniques that we hope will eventually apply to an arbitrary family of $L$-functions. This consists of a probability distribution on power series in $q^{-s}$ which combines properties of the characteristic polynomials of Haar-random unitary matrices and random Euler products over $\mathbb F_q[u]$. The support of our distribution is contained in the intersection of the supports of the two original distributions. The expectations of low-degree polynomials in the coefficients of our series approximate the expectations of the same polynomials in the coefficients of random Euler products, while the expectations of high-degree polynomials approximate the expectations of the same polynomials in the coefficients of the characteristic polynomials of random matrices. Furthermore, the expectations of absolute powers of our series approximate the Conrey-Farmer-Keating-Rubinstein-Snaith/Andrade-Keating prediction for the moments of our family of $L$-functions.