The distribution of random polynomials with multiplicative coefficients

Abstract: A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say $+1$ or $-1$ with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree $N$ polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large $N$). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund's results, exploring similarities and some differences.

Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.

This is joint work with Jacques Benatar and Alon Nishry.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

We will email out the link to all registered participants the day before.