Double Dirichlet series associated with arithmetic functions. II (Q6046577)

The general double Dirichlet series is defined by \[ \phi_{2}(s_1,s_2;\alpha_1,\alpha_2)=\sum_{m_1=1}^{+\infty}\sum_{m_2=1}^{+\infty}\frac{\alpha_{1}(m_1)\alpha_{2}(m_2)}{m_1^{s_1}(m_{1}+m_{2})^{s_2}}, \] where \(\alpha_1\) and \(\alpha_2\) are arithmetic functions. In this paper under review, the authors studied (see Theorem 3.1) the analytic behavior of \(\phi_{2}(s_1,s_2;1,\Lambda)\) at nonnegative integers points where \(\Lambda\) denotes the von Mangoldt function. Furthermore, in Theorem 5.1 they gives residues of \(\phi_{2}(s_1,s_2;1,\widetilde{\alpha})\) defined in Equation (1.5) for some pairs of integers \((s_1,s_2)\). In fact, this paper is a continuation of their previous work, Part I [Kodai Math. J. 44, No. 3, 437--456 (2021; Zbl 1495.11107)].