The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

TL;DR

This paper develops a hybrid Euler-Hadamard product formula for Dirichlet L-functions over function fields, proposes a splitting conjecture for moments, and provides partial proofs supporting it, extending the analogy of number field results.

Contribution

It introduces an exact Euler-Hadamard product formula in the function field setting and formulates a splitting conjecture for moments, supported by proofs for specific cases.

Findings

Main term of the Euler product moment explicitly obtained

Splitting conjecture supported for moments k=1,2

Provides a function field analogue of existing number field results

Abstract

For Dirichlet $L$-functions in $\mathbb{F}_q [T]$ we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the $2k$-th moment of the Dirichlet $L$-functions at $\frac{1}{2}$, averaged over primitive characters of modulus $R$, is asymptotic to (as $\mathrm{deg} R \longrightarrow \infty$) the $2k$-th moment of the Euler product multiplied by the $2k$-th moment of the Hadamard product. We explicitly obtain the main term of the $2k$-th moment of the Euler product, and we conjecture via random matrix theory the main term of the $2k$-th moment of the Hadamard product. With the splitting conjecture, this directly leads to a conjecture for the $2k$-th moment of Dirichlet $L$-functions. Finally, we lend support for the splitting conjecture by proving the cases $k=1,2$. This work is the function field analogue of the work of Bui and Keating. A notable difference in the function field setting is that the Euler-Hadamard product formula is exact, in that there is no error term.