The minimum modulus of gap power series and h-measure of exceptional sets

TL;DR

This paper investigates the asymptotic behavior of entire Dirichlet series, establishing conditions under which the series approximates a dominant term outside sets of finite h-measure, extending understanding of exceptional sets in complex analysis.

Contribution

It introduces new criteria for the asymptotic approximation of Dirichlet series outside sets with finite h-measure, generalizing previous results on exceptional sets.

Findings

Asymptotic relation holds outside sets of finite h-measure

Conditions on the growth of the Dirichlet series coefficients

Extension of classical results to broader classes of exceptional sets

Abstract

For entire Dirichlet series of the form $F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{z\lambda_n},\ 0\le\lambda_n\uparrow+\infty\ (n\to+\infty)$, we establish conditions under which the relation $$ F(x+iy)=(1+o(1))a_{\nu(x,F)}e^{(x+iy)\lambda_{\nu(x,F)}} $$ is true as $x\to+\infty$ outside some set $E$ such that $\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$ uniformly in $y\in\Bbb{R}$, where $h(x)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty)$ with non-decreasing to $+\infty$ derivative.