## Abstract

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.

### Similar content being viewed by others

## Data availability

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

## References

Erdös, P., Wintner, A.: Additive arithmetical functions and statistical independence. Am. J. Math.

**61**(3), 713–721 (1939)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)Kac, M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc.

**55**(7), 641–665 (1949)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)

Erdös, P., Rényi, A.: A probabilistic approach to problems of diophantine approximation. Ill. J. Math.

**1**(3), 303–315 (1957)Pólya, G.: Heuristic reasoning in the theory of numbers. Am. Math. Mon.

**66**(5), 375–384 (1959)Golomb, S.W.: A class of probability distributions on the integers. J. Number Theory

**2**(2), 189–192 (1970)NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, 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. https://dlmf.nist.gov/

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)

Diaconis, P., Erdös, P.: On the distribution of the greatest common divisor. Lecture Notes-Monograph Series

**45**, 56–61 (2004)Benkoski, S.J.: The probability that k positive integers are relatively r-prime. J. Number Theory

**8**(2), 218–223 (1976)Sittinger, B.D.: The probability that random algebraic integers are relatively r-prime. J. Number Theory

**130**(1), 164–171 (2010)Hu, J.: The probability that random positive integers are k-wise relatively prime. Int. J. Number Theory

**9**(5), 1263–1271 (2013)Ross, S.M.: A First Course in Probability, 10th edn. Pearson, Boston (2019)

## Author information

### Authors and Affiliations

### Corresponding author

## 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.

## About this article

### Cite this article

Fujita, T., Yoshida, N. On further application of the zeta distribution to number theory.
*Res. number theory* **9**, 81 (2023). https://doi.org/10.1007/s40993-023-00485-3

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/s40993-023-00485-3