  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.
