Dirichlet polynomials: some old and recent results, and their interplay in number theory

TL;DR

This paper explores Dirichlet polynomials' roles in classical number theory problems, analyzing their mean and supremum properties, and investigates their random counterparts using stochastic process methods.

Contribution

It provides a comprehensive review of Dirichlet polynomials' properties and introduces new stochastic analysis techniques for their supremum behavior.

Findings

Analysis of Dirichlet polynomials' mean properties

Investigation of supremum properties in deterministic and random cases

Application of stochastic process methods to number theory problems

Abstract

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is developed.