ENTIRE DIRICHLET SERIES AND h − MEASURE OF EXCEPTIONAL SETS - Скасків, Олег Богданович; Сало, Тетяна Михайлівна (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2019)
For entire Dirichlet series $F(z) =\sum_{n=0}^{+\infty} a_ne^{z\lambda_n},$ $0\le \lambda_n \uparrow +\infty$ $(n\to+\infty) ,$ we establish necessary conditions for the relation $\sup\{|F(x+iy)|: y \in\mathbb{R}\}= (1+o(1)) \max\{|a_n| e^{x\lambda_n}: n \ge 0\}$to hold when $x\to+\infty$ outside some set $E$ such that $\int_E dh (x) <+\infty,$ where $h$ is positive continuous function increasing to $+\infty$ on $[0;+\infty)$ with non-increasing derivative.