The multiple Dirichlet product and the multiple Dirichlet series

TL;DR

This paper introduces the multiple Dirichlet product, explores its properties, and derives a zero-free region and reciprocal expression for multiple Dirichlet series, advancing understanding of their analytic behavior.

Contribution

It defines the multiple Dirichlet product and derives new properties, including zero-free regions and reciprocal series expressions for multiple Dirichlet series.

Findings

Established zero-free regions for multiple Dirichlet series

Derived a series expression for the reciprocal of multiple Dirichlet series

Analyzed properties of the newly defined multiple Dirichlet product

Abstract

First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.