Minimum of the multiple Dirichlet series

Abstract

Conditions are established under which the following relation is satisfied:

$$M (x) = (1 + o (1)) m (x) = (1 + o (1)) \mu (x)$$

as $| x |\rightarrow + \infty$ outside a sufficiently small set, for an entire function $F (z)$ of several complex variables $z\in\mathbb{C}^p$, $p\geq 2$, represented by a Dirichlet series. Here $M (x) = \rm{sup}\{| F (x + iy)| : y \in \mathbb{R}^p\}$ and $m (x) =\rm{ inf} \{ | F (x + iy) | : y \in \mathbb{R}^p\}$, with $\mu (x)$ the maximal term of the Dirichlet series, $x \in \mathbb{R}^p$.