Behavior of entire Dirichlet series of class $\underline{D}(\Phi)$ on curves of bounded $K$-slope

Abstract

We study an asymptotic behavior of the sum of an entire Dirichlet series $F(s)=\sum\limits_{n}a_{n}e^{\lambda_{n}s}$, $0<\lambda_{n}\uparrow\infty$, on curves of a bounded $K$-slope naturally going to infinity. For entire transcendental functions of finite order having the form $f(z)=\sum\limits_{n}a_{n}z^{p_{n}}$, $p_{n}\in\mathbb{N}$, Pólya showed that if the density of the sequence $\left\{p_{n}\right\}$ is zero, then for each curve $\gamma$ going to infinity there exists an unbounded sequence $\{\xi_{n}\}\subset\gamma$ such that, as $\xi_{n}\rightarrow\infty$, the relation holds: \begin{equation*} \ln M_{f}(|\xi_{n}|)\sim \ln\left|f(\xi_{n})\right|; \end{equation*} here $M_{f}(r)$ is the maximum of the absolute value of the function $f$. Later these results were completely extended by I.D. Latypov to entire Dirichlet series of finite order and finite lower order according in the Ritt sense. A further generalization was obtained in works by N.N. Yusupova–Aitkuzhina to more general classes $D(\Phi)$ and $\underline{D}(\Phi)$ defined by the convex majorant $\Phi$. In this paper we obtain necessary and sufficient conditions for the exponents $\lambda_{n}$ ensuring that the logarithm of the absolute value of the sum of any Dirichlet series from the class $\underline{D}(\Phi)$ on the curve $\gamma$ of a bounded $K$-slope is equivalent to the logarithm of the maximum term as $\sigma=\mathrm{Re}\, s\rightarrow +\infty$ over some asymptotic set, the upper density of which is one. We note that for entire Dirichlet series of an arbitrarily fast growth the corresponding result for the case of $\gamma =\mathbb{R}_+$ was obtained by A.M. Gaisin in 1998.