Universality of $L$-Functions over function fields

TL;DR

This paper proves that Dirichlet L-functions over function fields are universal, meaning they can approximate any nonvanishing analytic function arbitrarily closely when the modulus degree is sufficiently high.

Contribution

It establishes the universality property of Dirichlet L-functions over function fields, extending the understanding of their approximation capabilities.

Findings

L-functions are universal over function fields

Approximate any nonvanishing analytic function arbitrarily closely

Universality holds for high enough degree moduli

Abstract

We prove that the Dirichlet $L$-functions associated with Dirichlet characters in $\mathbb{F}_{q}[x]$ are universal. That is, given a modulus of high enough degree, $L$-functions with characters to this modulus can be found that approximate any given nonvanishing analytic function arbitrarily closely.