Comparative growth of an entire function and the integrated counting function of its zeros

Abstract

Let $(\zeta_n)$ be a sequence of complex numbers such that $\zeta_n\to\infty$ as $n\to\infty$, $N(r)$ be the integrated counting function of this sequence, and let $\alpha$ be a positive continuous and increasing to $+\infty$ function on $\mathbb{R}$ for which $\alpha(r)=o(\log (N(r)/\log r))$ as $r\to+\infty$. It is proved that for any set $E\subset(1,+\infty)$ satisfying $\int_{E}r^{\alpha(r)}dr=+\infty$, there exists an entire function $f$ whose zeros are precisely the $\zeta_n$, with multiplicities taken into account, such that the relation $$ \liminf_{r\in E,\ r\to+\infty}\frac{\log\log M(r)}{\log r\log (N(r)/\log r)}=0 $$ holds, where $M(r)$ is the maximum modulus of the function $f$. It is also shown that this relation is best possible in a certain sense.