Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type - Salo, T. M.; Skaskiv, O. B. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
For entire Dirichlet series of the form \(F\left( z \right) = \sum olimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }\) , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{ u }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{ u }\left( {\sigma } \right)} }$$ 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, +∞).