On Sums of Zeros of Infinity Order Entire Functions

Abstract

We consider an infinite order entire functions $f(z)$, whose zeros $z_1(f), z_2(f),\dots$ are enumerated in the increasing order. For a nondecreasing sequence $\{p_k\}$ of positive numbers, a bound for the sums $$ \sum_{k=1}^j \frac{1}{|z_k(f)|^{p_k}}\;\;(j=1, 2,\dots) $$ is suggested. That bound gives us conditions providing the convergence of the corresponding series.