On the domain of convergence of general Dirichlet series with complex exponents

Abstract

Let $(\lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=\sum\limits_{n=0}^{+\infty} a_ne^{z\lambda_n}$, $z\in\mathbb{C}$, we denote $G_{\mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $\mu(z,F)=\max\big\{|a_n|e^{\Re(z\lambda_n)} : n\geq 0\big\}$, respectively. It is well known that $G_\mu(F), G_a(F)$ are convex domains.

Let us denote $\mathcal{N}_1(z):=\{n : \Re(z\lambda_n)>0\}$, $\mathcal{N}_2(z):=\{n : \Re(z\lambda_n)<0\}$ and \[\alpha^{(1)}(\theta) :=\varliminf\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_1(e^{i\theta})}}\frac{-\ln|a_n|}{\Re(e^{i\theta}\lambda_n)},\qquad \alpha^{(2)}(\theta) :=\varlimsup\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_2(e^{i\theta})}}\frac{-\ln|a_n|}{\Re(e^{i\theta}\lambda_n)}.\] Assume that $a_n\to 0$ as $n\to +\infty$. In the article, we prove the following statements.

$1)$ If $\alpha^{(2)}(\theta)<\alpha^{(1)}(\theta)$ for some $\theta\in [0,\pi)$ then \[\big\{te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\big\}\subset G_\mu(F)\] as well as \[\big\{te^{i\theta} : t\in (-\infty,\alpha^{(2)}(\theta))\cup (\alpha^{(1)}(\theta),+\infty)\big\}\cap G_\mu(F)=\emptyset.\]

$2)$ $G_\mu(F)=\bigcup\limits_{\theta\in [0,\pi)}\{z=te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\}.$

$3)$ If $h:=\varliminf\limits_{n\to +\infty}\frac{-\ln |a_n|}{\ln n}\in (1,+\infty)$, then \[\Big(\frac{h}{h-1}\cdot G_a(F)\Big)\supset G_\mu(F)\supset G_c(F).\] If $h=+\infty$ then $G_a(F)=G_c(F)=G_\mu(F)$, therefore $G_c(F)$ is also a convex domain.