On the abscises of the convergence of multiple Dirichlet series

• Main Authors: O. Yu. Zadorozhna, O. B. Skaskiv
• Format: Article
• Language: English
• Published: Vasyl Stefanyk Precarpathian National University 2013-01-01
• Series: Karpatsʹkì Matematičnì Publìkacìï
• Online Access: http://journals.pu.if.ua/index.php/cmp/article/view/26

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 $e_{1}=(1,...,1)\in \mathbb{R}^p,\;\; \delta_0\in \mathbb{R},$ by condition $\liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+\delta_0\|\lambda_{(n)}\|}{\ln\|n\|}>p;$ $\gamma G_c\subset G_a+\delta; \;\; \gamma G_{\mu}\subset G_a+\delta,$ where $\delta\in\mathbb{R}^{p},$ by condition $\liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+(\delta,\lambda_{(n)})}{\ln\,n_1+...+\ln\,n_p}>1.$