Summary: We study the behavior of zero-sets of the double zeta-function \(\zeta_2(s_1,s_2)\) (and also of more general multiple zeta-function \(\zeta_r(s_1,\dots, s_r))\). In our earlier paper we studied the case \(s_1=s_2\), while in the present paper we consider a more general two-variables situation. We carry out numerical computations in order to trace the behavior of zero-sets of \(\zeta_2(s_1, s_2)\). We observe that some zero-sets approach the points \((s_1, s_2)\) with \(s_2=0\), while other zero-sets approach the points \((s_1, s_2)\) with \(s_2\) being solutions of \(\zeta(s_2)=1\). In the former case, when \(s_2\) tends to 0, we observe that \(\text{Im} s_1\) comes close to the imaginary part of a non-trivial zero of the Riemann zeta-function. In the latter case we give a theoretical proof, in the general \(r\)-fold setting.
For Part I see [the authors, Mosc. J. Comb. Number Theory 4, No. 3, 21–39 (2014; Zbl 1352.11082)].