The average value of a certain number-theoretic function over the primes

Louis Rubin
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 564–570
DOI: 10.7546/nntdm.2023.29.3.564-570
Full paper (PDF, 212 Kb)


Authors and affiliations

Louis Rubin
Department of Mathematics, Florida State University
Tallahassee, Florida, United States


We consider functions F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0} for which there exists a positive integer n such that two conditions hold: F(p) divides n for every prime p, and for each divisor d of n and every prime p, we have that d divides F(p) iff d divides F(p\textnormal{ mod }d). Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an F over all primes p, recovering a theorem of these authors as a special case. As an application, we compute the average number of r-periodic points of a multivariate power map defined on a product Z_{f_1(p)}\times\cdots\times Z_{f_m(p)} of cyclic groups, where f_i(t) is a polynomial.


  • Average value
  • Prime number
  • Periodic points
  • Cyclic groups

2020 Mathematics Subject Classification

  • 37C25
  • 11N37


  1. Khrennikov, A., & Nilsson, M. (2001). On the number of cycles of p-adic dynamical systems. Journal of Number Theory, 90, 255–264.
  2. Oleschko, K., Khrennikov, A., Oleshko, B., & Parrot, J. (2017). The primes are everywhere, but nowhere… New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences, Springer, 155–167.

Manuscript history

  • Received: 18 April 2023
  • Revised: 21 July 2023
  • Accepted: 26 July 2023
  • Online First: 2 August 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Rubin, L. (2023). The average value of a certain number-theoretic function over the primes. Notes on Number Theory and Discrete Mathematics, 29(3), 564-570, DOI: 10.7546/nntdm.2023.29.3.564-570.

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