This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius 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.

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