The growth of Weierstrass canonical products of genus zero with random zeros

Abstract

Let $\zeta=(\zeta_n)$ be a complex sequence of genus zero, $\tau$ be its exponent of convergence, $N(r)$ be its integrated counting function, $\pi(z)=\prod\bigl(1-\frac{z}{\zeta_n}\bigr)$ be the Weierstrass canonical product, and $M(r)$ be the maximum modulus of this product. Then, as is known, the Wahlund-Valiron inequality
$$
\limsup_{r\to+\infty}\frac{N(r)}{\ln M(r)}\ge w(\tau),\qquad w(\tau):=\frac{\sin\pi\tau}{\pi\tau},
$$
holds, and this inequality is sharp. It is proved that for the majority (in the probability sense) of sequences $\zeta$ the constant $w(\tau)$ can be replaced by the constant $w\left(\frac{\tau}2\right)$ in the Wahlund-Valiron inequality.