Double Dirichlet series associated with arithmetic functions II

TL;DR

This paper extends previous work on double Dirichlet series linked to arithmetic functions, analyzing their behavior near singularities and establishing a reciprocity law for their residues, while correcting earlier inaccuracies.

Contribution

It introduces new results on the analytic behavior of double Dirichlet series at non-positive integers and proves a reciprocity law for their residues, correcting prior errors.

Findings

Residue reciprocity law established

Behavior near non-positive integers analyzed

Corrections made to previous results

Abstract

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the M\"obius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.