Number-theoretic functions for Gaussian integers

Authors

DOI:

https://doi.org/10.17721/1812-5409.2023/1.1

Keywords:

Gaussian integers, divisor, number of divisors, sum of the divisors, product of the divisors

Abstract

The classical number-theoretic functions – a number of divisors τ(n), sum of the divisors σ(n) and product of the divisors π(n) of a positive integer n – were generalized to the ring Z[i] of Gaussian integers. For the evaluation of the corresponding functions τ*(α), σ*m(α) and π*(α), obtained were the explicit formulae that use the canonical representation of α. A number of properties of these functions were studied, in particular, estimates from above for the functions τ*(α) and σ*m(α) and the properties connected with divisibility of their values by certain numbers. Researched are also sums of products of powers of the divisors for α∈Z[i].

Pages of the article in the issue: 11 - 20

Language of the article: English

References

STILLWELL, J. (2010) Mathematics and Its History. New York: Undergraduate Texts in Mathematics, Springer Science+Business Media.

GAUSS, C. F. (1973) Theoria residuorum biquadraticorum. Commentatio secunda. Comm. Soc. Reg. Sci. G¨ottingen 7: 89-148; reprinted in Werke, Hildesheim: Georg Olms Verlag. p. 93–148.

CALCUT, J. S.(2009) Gaussian Integers and Arctangent Identities for π. The American Mathematical Monthly. V. 116(6). p. 515-530

STILLWELL, J. (2003) Elements of Number Theory. New York: Springer Science+Business Media.

WILLERDING, M. F. (1966) Divisibility and factorization of Gaussian integers. The Mathematics Teacher. V. 59(7). p. 634-637.

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Published

2023-07-13

How to Cite

Arskyi, N. (2023). Number-theoretic functions for Gaussian integers. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (1), 11–20. https://doi.org/10.17721/1812-5409.2023/1.1

Issue

No. 1 (2023)

Section

Algebra, Geometry and Probability Theory

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Copyright (c) 2023 Nikita Arskyi

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