Absolute convergence of general multiple Dirichlet series (Q6159118)

In this paper, the author studies the absolute convergence of general multiple Dirichlet series defined by \[ \Phi_r((s_j); (a_j))= \sum_{m_1=1}^\infty\sum_{m_2=1}^\infty\cdots\sum_{m_r=1}^\infty\frac{a_1(m_1)a_2(m_2)\cdots a_r(m_r)}{m_1^{s_1}(m_1+m_2)^{s_2}\cdots (m_1+m_2+\cdots+m_r)^{s_r}}, \] where \(a_i\) are arithmetic functions. The function \(\Phi_r((s_j); (a_j))\) is one of the generalizations of the multiple zeta function defined by \[ \sum_{m_1=1}^\infty\sum_{m_2=1}^\infty\cdots\sum_{m_r=1}^\infty\frac{1}{m_1^{s_1}(m_1+m_2)^{s_2}\cdots (m_1+m_2+\cdots+m_r)^{s_r}}. \] Therefore, it is a natural question to find the region of absolute convergence of the multiple Dirichlet series \(\Phi_r((s_j); (a_j))\) as in the multiple zeta function. The author first shows the following result. We assume that for each \(j\), the series \(\displaystyle\sum_{m=1}^{\infty}\frac{a_j(m)}{m^s}\) converges absolutely for \(\Re(s)>\alpha_j>0\). Then the multiple Dirichlet series \(\Phi_r((s_j); (a_j))\) converges absolutely in the region \[ \{(s_1,\ldots,s_r)~|~\Re(s_r+\cdots+s_{r-i})>\alpha_r+\cdots+\alpha_{r-i}\text{ for }0\le i\le r-1\}. \] Under a certain condition on the arithmetic functions, the author next proves the following stronger result. We now assume that for each \(j\), the arithmetic functions \(a_j(m)\) and positive real numbers \(\alpha_j\) satisfy the following conditions \begin{itemize} \item the series \(\displaystyle\sum_{m=1}^{\infty}\frac{a_j(m)}{m^s}\) has abscissa of absolute convergence \(\alpha_j\), \item the sum \(\displaystyle\sum_{m\le t}|a_j(m)|\gg t^{\alpha_j}\) for every \(t\ge1\). \end{itemize} Then, the multiple Dirichlet series \(\Phi_r((s_j); (a_j))\) converges absolutely at \((s_1,\ldots,s_r)\) if and only if \(\Re(s_r+\cdots+s_{r-i})>\alpha_r+\cdots+\alpha_{r-i}\) for \(0\le i\le r-1\). The author finally shows that if the multiple Dirichlet series \(\Phi_r((s_j); (a_j))\) converges absolutely at \((s_1',\ldots,s_r')\), then \(\Phi_r((s_j); (a_j))\) converges absolutely at each \((s_1,\ldots,s_r)\) with \(\Re(s_r+\cdots+s_{r-i})>\Re(s'_r+\cdots+s'_{r-i})\) for \(0\le i\le r-1\).