Supremum of Random Dirichlet Polynomials with Sub-multiplicative Coefficients

TL;DR

This paper investigates the maximum size of random Dirichlet polynomials with sub-multiplicative coefficients using Gaussian process comparison, providing new insights without relying on metric entropy methods.

Contribution

It introduces a Gaussian process approach to analyze the supremum of random Dirichlet polynomials with sub-multiplicative coefficients, avoiding traditional metric entropy techniques.

Findings

Derived bounds for the supremum of the polynomials.

Established comparison principles for Gaussian processes in this context.

Provided a novel methodological framework for similar problems.

Abstract

We study the supremum of random Dirichlet polynomials $D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $ d $ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.