Almost periodic stochastic processes with applications to analytic number theory

TL;DR

This paper extends classical results on the asymptotic distribution of almost periodic functions to stochastic processes, providing new insights and applications in analytic number theory.

Contribution

It introduces a functional extension of asymptotic distribution results for Besicovitch almost periodic functions to stochastic processes, with applications to number theory.

Findings

Convergence in distribution of stochastic processes derived from almost periodic functions.

Characterization of the limiting stationary process.

Applications to error terms in prime number theory.

Abstract

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly distributed on $[-1,1]$, then the random variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.