On the abscises of the convergence of multiple Dirichlet series

Abstract

For multiple Dirichlet series of the form $F(s)=\sum_{\|n\|=0}^\infty a_{(n)}\exp\{(\lambda_{(n)},s)\}$ we establish relations between domains of the convergence $G_c$, absolutely convergence $G_a$ and of the domain of the existence of the maximal term $G_{\mu}$ of the series as follows: $\gamma G_{c}\subset G_{a}+\delta_0 e_{1},\ \gamma G_{\mu}\subset G_{a}+\delta_0 e_{1},$ where $e_{1}=(1,\dots,1)\in \mathbb{R}^p, \;\; \delta_0\in \mathbb{R},$ by condition $ \varliminf\limits_{\|n\|\to\infty} \frac{(\gamma-1)\ln\,|a_{(n)}|+\delta_0\|\lambda_{(n)}\|}{\ln\|n\|}>p;$ $\gamma G_c\subset G_a+\delta; \;\; \gamma G_{\mu}\subset G_a+\delta,$ where $\delta\in\mathbb{R}^{p},$ by condition $\varliminf\limits_{\|n\|\to\infty} \frac{(\gamma-1)\ln\,|a_{(n)}|+(\delta,\lambda_{(n)})}{\ln\,n_1+...+\ln\,n_p}>1.$