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

Abstract

Let $zeta=(zeta_n)$ be a complex sequence of genus zero, $au$ be its exponent ofconvergence, $N(r)$ be its integrated counting function,$pi(z)=prodigl(1-frac{z}{zeta_n}igr)$ be the Weierstrass canonical product, and$M(r)$ be the maximum modulus of this product. Then, as is known, the Wahlund-Valironinequality$$varlimsup_{ro+infty}frac{N(r)}{ln M(r)}ge w(au),qquad w(au):=frac{sinpiau}{piau},$$holds, and this inequality is sharp. It is proved that for the majority (in theprobability sense) of sequences $zeta$ the constant $w(au)$ can be replacedby the constant $wleft(frac{au}2ight)$ in the Wahlund-Valironinequality.

Keywords

• entire function
• Weierstrass products
• maximum modulus
• order
• genus
• exponent of convergence
• integrated counting function