Skip to main content
Log in

On further application of the zeta distribution to number theory

  • Research
  • Published:
Research in Number Theory Aims and scope Submit manuscript


This note gives some applications of the zeta distribution to number theory. First, a number of probabilistic proofs of the equality between the Riemann zeta functions and some number-theoretic functions are provided. Second, some problems of probability with respect to positive integers considered in the literature are discussed in the zeta distribution case. In contrast to usual heuristic arguments of probabilistic approach in number theory where the uniform distribution is assumed on positive integers, the proofs here are rigorous by utilizing the zeta distributions on positive integers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Data availability

Data sharing not applicable because no datasets were analysed during the current study.


  1. Erdös, P., Wintner, A.: Additive arithmetical functions and statistical independence. Am. J. Math. 61(3), 713–721 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  2. Erdös, P., Kac, M.: The gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62(1), 738–742 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kac, M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55(7), 641–665 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  4. Erdös, P.: On additive arithmetical functions and applications of probability to number theory. In: Proceedings of the International Congress of Mathematicians, Amsterdam, vol. III,, pp. 13–19 (1954)

  5. Erdös, P., Rényi, A.: A probabilistic approach to problems of diophantine approximation. Ill. J. Math. 1(3), 303–315 (1957)

    MathSciNet  MATH  Google Scholar 

  6. Pólya, G.: Heuristic reasoning in the theory of numbers. Am. Math. Mon. 66(5), 375–384 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golomb, S.W.: A class of probability distributions on the integers. J. Number Theory 2(2), 189–192 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. NIST Digital Library of Mathematical Functions., Release 1.1.9 of 2023-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

  9. Abramowitz, M., Stegun, I.A., Romer, R.H.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. American Association of Physics Teachers, New York (1988)

    Book  MATH  Google Scholar 

  10. Diaconis, P., Erdös, P.: On the distribution of the greatest common divisor. Lecture Notes-Monograph Series 45, 56–61 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benkoski, S.J.: The probability that k positive integers are relatively r-prime. J. Number Theory 8(2), 218–223 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sittinger, B.D.: The probability that random algebraic integers are relatively r-prime. J. Number Theory 130(1), 164–171 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, J.: The probability that random positive integers are k-wise relatively prime. Int. J. Number Theory 9(5), 1263–1271 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ross, S.M.: A First Course in Probability, 10th edn. Pearson, Boston (2019)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Naohiro Yoshida.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fujita, T., Yoshida, N. On further application of the zeta distribution to number theory. Res. number theory 9, 81 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


MSC Classification: