Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type

Abstract

For entire Dirichlet series of the form $F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }$, we establish conditions under which the relation

$$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$

holds uniformly in $y \in \mathbb{R}\;{as}\;{\sigma } \to + \infty$ outside a certain set E for which

$$DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$$

where h(σ) is a positive continuous function increasing to +∞ on [0, +∞).