Sums of $L$-functions over the rational function field

TL;DR

This paper explicitly computes certain multiple Dirichlet series over the rational function field, revealing their structure through functional equations and a simple correspondence with their local components.

Contribution

It provides explicit formulas for multiple Dirichlet series related to quadratic and higher-order $L$-functions over the rational function field, using functional equations.

Findings

Explicit formulas for the multiple Dirichlet series.

Identification of a simple correspondence with local $p$-parts.

Utilization of functional equations to compute series.

Abstract

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order $L$-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the $L$-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their $p$-parts.