Growth of entire functions represented by Dirichlet series

V. A. Oskolkov ; L. I. Kalinichenko

Sbornik. Mathematics, Tome 187 (1996) no. 10, pp. 1545-1560

Résumé

Let, $\displaystyle F(z)=\sum _{n=1}^\infty a_ne^{\lambda _nz}$ be an entire function represented in the whole of the plane by an absolutely convergent Dirichlet series such that $$ 0\leqslant \lambda _1\lambda _2\dotsb ,\qquad \varlimsup _{n\to \infty }\frac {\ln n}{\lambda _n}=\mu \in [0,+\infty ). $$ The connection between the growth of the quantity $$ M(F;x)=\sup \bigl \{|F(x+iy)|:|y|+\infty \bigr \},\qquad x\to +\infty. $$ End the behaviour of $|a_n|$ and $\lambda_n$ as $n\to \infty$ is described in general form.