Distribution of zeroes of Rademacher Taylor series

TL;DR

This paper analyzes the asymptotic distribution of zeros of Rademacher Taylor series, providing detailed counting functions and addressing open questions from classical work, using advanced Fourier series techniques.

Contribution

It introduces new asymptotic formulas for zero distributions of Rademacher Taylor series, extending classical results and employing recent Fourier series integrability findings.

Findings

Asymptotics of zero counting function derived

Asymptotics of weighted zero counting function obtained

Answers to open questions from 1948 work

Abstract

We find the asymptotics of the counting function of zeroes of random entire functions represented by Rademacher Taylor series. We also give the asymptotics of the weighted counting function, which takes into account the arguments of zeroes. These results answer several questions left open after the pioneering work of Littlewood and Offord of 1948. The proofs are based on our recent result on the logarithmic integrability of Rademacher Fourier series.