{
  "id": "2502.04969",
  "version": "v1",
  "published": "2025-02-07T14:36:29.000Z",
  "updated": "2025-02-07T14:36:29.000Z",
  "title": "Almost periodic stochastic processes with applications to analytic number theory",
  "authors": [
    "Alexander Iksanov",
    "Zakhar Kabluchko",
    "Alexander Marynych"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-07T14:36:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "60F17",
      "60G10",
      "11M26"
    ],
    "keywords": [
      "analytic number theory",
      "periodic stochastic processes",
      "periodic function",
      "applications",
      "prime number theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}