Asymptotics of the entire functions with $\upsilon$-density of zeros along the logarithmic spirals

Abstract

Let $\upsilon$ be the growth function such that $r\upsilon'(r)/\upsilon (r) \to 0$ as $r \to +\infty$, $l_\varphi^c = \{z=te^{i(\varphi+c \ln t)}, 1 \leqslant t < +\infty\}$ be the logarithmic spiral, $f$ be the entire function of zero order. The asymptotics of $\ln f(re^{i(\theta +c \ln r)})$ along ordinary logarithmic spirals $l_\theta^c$ of the function $f$ with $\upsilon$-density of zeros along $l_\varphi^c$ outside the $C_0$-set is found. The inverse statement is true just in case zeros of $f$ are placed on the finite logarithmic spirals system $\Gamma_m = \bigcup_{j=0}^m l_{\theta_j}^c$.