Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents

Abstract

It is proved analogues of the classical Wiman's inequality} for  the class $\mathcal{D}$ of absolutely convergents in the whole complex plane $\mathbb{C}^p$ (entire) Dirichlet series of the form $\displaystyle F(z)=\sum\limits_{\|n\|=0}^{+\infty} a_ne^{(z,\lambda_n)}$ with such a sequence of exponents $(\lambda_n)$ that $\{\lambda_n\colon n\in\mathbb{Z}^p\}\subset \mathbb{C}^p$ and $\lambda_n\not=\lambda_m$ for all $n\not= m$. For $F\in\mathcal{D}$ and $z\in\mathbb{C}^p\setminus\{0\}$ we denote  $\mathfrak{M}(z,F):=\sum\limits_{\|n\|=0}^{+\infty}|a_n|e^{\Re(z,\lambda_n)},\quad\mu(z,F):=\sup\{|a_n|e^{\mathop{\rm Re}(z,\lambda_n)}\colon n\in\mathbb{Z}^ p_+\},$ $(m_k)_{k\geq 0}$ is $(\mu_{k})_{k\geq 0}$ the sequence $(-\ln|a_{n}|)_{n\in\mathbb{Z}^p_+}$ arranged by non-decreasing. The  main result of the paper: Let $F\in \mathcal{D}.$ If $(\exists \alpha > 0)\colon$ $\int\nolimits_{t_0}^{+\infty}t^{-2}{(n_1(t))^{\alpha}}dt<+\infty,$  $n_1(t)\overset{def}=\sum\nolimits_{\mu_n\leq t} 1,\quad t_0>0,$ then there exists a set $E\subset\gamma_{+}(F),$\ such that $\tau_{2p}(E\cap\gamma_{+}(F))=\int_{E\cap\gamma_{+}(F)}|z|^{-2p}dxdy\leq C_p, z=x+iy\in\mathbb{C}^p,$  and relation $\mathfrak{M}(z,F)= o(\mu(z,F)\ln^{1/\alpha} \mu(z,F))$ holds as $z\to \infty$\ $(z\in \gamma_R\setminus E)$ for each $R>0$, where $\gamma_R=\Big\{z\in\mathbb{C}^p\setminus\{0\}\colon\ K_F(z)\leq R \Big\},\quad K_F(z)=\sup\Big\{\frac1{\Phi_z( t)}\int^{ t}_0 \frac {{\Phi_z}(u)}{u} du\colon\ t \geq t_0\Big\},$ $\gamma(F)=\{z\in\mathbb{C}\colon \ \lim\limits_{t\to +\infty}\Phi_z(t)=+\infty\},\quad \gamma_+(F)=\mathop{\cup}_{R>0}\gamma_R$, $\Phi_z(t)=\frac1{t}\ln\mu(tz,F)$. In general, under the specified conditions, the obtained inequality is exact.