# Multiple Dirichlet series associated to additive and Dirichlet characters

## TL;DR

This paper investigates the analytic properties of multiple Dirichlet series linked to additive and Dirichlet characters, establishing meromorphic continuation and singularity sets for additive cases, and exploring connections for Dirichlet cases.

## Contribution

It provides the first meromorphic continuation for multiple Dirichlet series associated to additive characters and analyzes their singularities, also relating these to Dirichlet characters.

## Key findings

- Meromorphic continuation for additive character series
- Exact singularity set for additive character series
- Connection between additive and Dirichlet character series

## Abstract

In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established via obtaining translation formulas satisfied by these multiple Dirichlet series. While it seems difficult to obtain such a translation formula for the multiple Dirichlet series associated to Dirichlet characters, we rely on their intrinsic connection with the multiple Dirichlet series associated to additive characters in order to investigate their analytic characteristics. We are also able to determine the exact set of singularities of the multiple Dirichlet series associated to additive characters.