Double Dirichlet series associated with arithmetic functions

TL;DR

This paper studies double Dirichlet series linked to arithmetic functions, demonstrating their analytic continuation via Mellin-Barnes integrals and examining their values at non-positive integers.

Contribution

It introduces a method for analytic continuation of double Dirichlet series associated with key arithmetic functions using Mellin-Barnes integrals.

Findings

Analytic continuation of double Dirichlet series achieved

Values at non-positive integers observed and analyzed

Method applicable to various arithmetic functions

Abstract

We consider double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the M\"obius function, and so on. We show analytic continuations of them by use of the Mellin-Barnes integral. Furthermore we observe their reverse values at non-positive integer points.