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Almost periodic stochastic processes with applications to analytic number theory

Alexander Iksanov, Zakhar Kabluchko, Alexander Marynych

TL;DR

This work develops a functional limit theory for Besicovitch almost periodic functions under random translations, establishing convergence in the Besicovitch space $B_2(\mathcal{S}(f))$ to a limiting stationary random process $\mathcal{M}_f$ with a discrete spectrum. The limit admits a Fourier-series description with random coefficients, and the authors analyze its continuity, ergodicity, and tangent processes, including a Brownian-type limit under regular variation of spectral mass. They further connect these probabilistic limit objects to analytic number theory by deriving functional limit results for classical arithmetic functions such as von Mangoldt’s $\psi$, Möbius/Mertens $M$ and $L$, under RH and related conjectures (Gonek-type hypotheses). These results provide a probabilistic lens on error terms and fluctuations intimately tied to zeros of the Riemann zeta-function, offering a bridge between almost periodic analysis and number-theoretic asymptotics. The paper also proposes conjectures on lifting finite-dimensional convergence to functional convergence in spaces of continuous paths, highlighting potential broader applicability to number-theoretic problems.

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.

Almost periodic stochastic processes with applications to analytic number theory

TL;DR

This work develops a functional limit theory for Besicovitch almost periodic functions under random translations, establishing convergence in the Besicovitch space to a limiting stationary random process with a discrete spectrum. The limit admits a Fourier-series description with random coefficients, and the authors analyze its continuity, ergodicity, and tangent processes, including a Brownian-type limit under regular variation of spectral mass. They further connect these probabilistic limit objects to analytic number theory by deriving functional limit results for classical arithmetic functions such as von Mangoldt’s , Möbius/Mertens and , under RH and related conjectures (Gonek-type hypotheses). These results provide a probabilistic lens on error terms and fluctuations intimately tied to zeros of the Riemann zeta-function, offering a bridge between almost periodic analysis and number-theoretic asymptotics. The paper also proposes conjectures on lifting finite-dimensional convergence to functional convergence in spaces of continuous paths, highlighting potential broader applicability to number-theoretic problems.

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 is a Besicovitch almost periodic function and is a random variable uniformly distributed on , then the random variables converge in distribution, as , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes 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.

Paper Structure

This paper contains 16 sections, 17 theorems, 143 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A brief reminder on almost periodic functions
  3. Besicovitch almost periodic functions
  4. Mean values and Fourier series
  5. Besicovitch functions with a restricted spectrum
  6. Almost periodic stochastic processes
  7. Convergence in distribution on B2
  8. The limit process for randomly translated almost periodic function
  9. Continuity of the limit process
  10. The tangent process
  11. Locally uniform convergence: a conjecture
  12. Applications in analytic number theory
  13. The von Mangoldt function and its partial sums
  14. The Möbius and Mertens functions
  15. The Liouville function and its partial sums
  16. ...and 1 more sections

Key Result

Proposition 1

For every $A\subseteq\mathbb{R}$, $B_2(A)=\{f\in B_2:\,\mathcal{S}(f)\subseteq A\}$.

Figures (2)

  • Figure 1: Realisation of the process $\mathcal{M}_{\psi_{ap}}(t)=2{\rm Re}(\sum_{k\geq 1}\rho_k^{-1}\mathcal{U}_k{\rm e}^{{\rm i} \gamma_k t})$, $t\in [0,10]$, appearing in Section \ref{['subsec:mangoldt']}.
  • Figure 2: Correlation function of the process $\mathcal{M}_{\psi_{ap}}$ appearing in Section \ref{['subsec:mangoldt']}, for $t\in [0,10]$.

Theorems & Definitions (33)

  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 1
  • proof
  • Theorem 3.3
  • ...and 23 more