The growth of Weierstrass canonical products of genus zero with random zeros - Zakharko, Yu.B.; Filevych, P.V. (Carpathian Mathematical Publications, 2013)
Let $ζ=(ζ_n)$ be a complex sequence of genus zero, $τ$ be its exponent of convergence, $N(r)$ be its integrated counting function, $π(z)=\prod\bigl(1-\frac{z}{ζ_n}\bigr)$ be the Weierstrass canonical product, and $M(r)$ be the maximum modulus of this product. Then, as is known, the Wahlund-Valiron inequality $$ \limsup_{r→+∞}\frac{N(r)}{\ln M(r)}\ge w(τ),\qquad w(τ):=\frac{\sinπτ}{πτ}, $$ holds, and this inequality is sharp. It is proved that for the majority (in the probability sense) of sequences $ζ$ the constant $w(τ)$ can be replaced by the constant $w\left(\frac{τ}2\right)$ in the Wahlund-Valiron inequality.