Involve: A Journal of Mathematics

  • Involve
  • Volume 9, Number 1 (2016), 27-40.

On the distribution of the greatest common divisor of Gaussian integers

Tai-Danae Bradley, Yin Choi Cheng, and Yan Fei Luo

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm x or less as x goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdős.

Article information

Source
Involve, Volume 9, Number 1 (2016), 27-40.

Dates
Received: 27 March 2013
Revised: 9 January 2015
Accepted: 28 January 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370969

Digital Object Identifier
doi:10.2140/involve.2016.9.27

Mathematical Reviews number (MathSciNet)
MR3438443

Zentralblatt MATH identifier
1330.11058

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 11K65: Arithmetic functions [See also 11Nxx] 60E05: Distributions: general theory

Keywords
Gaussian integer gcd moment Dedekind zeta function

Citation

Bradley, Tai-Danae; Cheng, Yin Choi; Luo, Yan Fei. On the distribution of the greatest common divisor of Gaussian integers. Involve 9 (2016), no. 1, 27--40. doi:10.2140/involve.2016.9.27. https://projecteuclid.org/euclid.involve/1511370969


Export citation

References

  • E. Cesàro, Ann. Mat. Pura Appl. 13:2 (1885), 233–268.
  • J. Christopher, “The asymptotic density of some $k$-dimensional sets”, Amer. Math. Monthly 63 (1956), 399–401.
  • G. E. Collins and J. R. Johnson, “The probability of relative primality of Gaussian integers”, pp. 252–258 in Symbolic and algebraic computation (Rome, 1988), edited by P. Gianni, Lecture Notes in Comput. Sci. 358, Springer, Berlin, 1989.
  • P. Diaconis and P. Erdős, “On the distribution of the greatest common divisor”, pp. 56–61 in A festschrift for Herman Rubin, edited by A. DasGupta, IMS Lecture Notes Monogr. Ser. 45, Inst. Math. Statist., Beachwood, OH, 2004.
  • S. R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications 94, Cambridge University Press, 2003.
  • M. N. Huxley, “Exponential sums and lattice points, III”, Proc. London Math. Soc. $(3)$ 87:3 (2003), 591–609.
  • F. Mertens, “Ueber einige asymptotische Gesetze der Zahlentheorie”, J. Reine Angew. Math. 77 (1874), 289–338.
  • G. Micheli and A. Ferraguti, “On Mertens–Cesáro theorem for number fields”, preprint, 2015. To appear in Bull. Aust. Math. Soc.
  • G. Micheli and R. Schnyder, “On the density of coprime $m$-tuples over holomorphy rings”, Int. J. Number Theory (online publication September 2015).
  • A. Schinzel, “Wacław Sierpiński's papers on the theory of numbers”, Acta Arith. 21 (1972), 7–13. (errata insert).
  • W. Sierpiński, “O pewnem zagadnieniu z rachunku funkcyj asymptotycznych”, Prace Matematyczno-Fizyczne 17 (1906), 77–118.
  • W. Sierpiński, “O sumowaniu szeregu $\sum_{n>a}^{n\leq b}{\tau(n)f(n)}$, gdzie $\tau(n)$ oznacza liczbę rozkładów liczby $n$ na sumę kwadratów dwóch liczb całkowitych”, Prace Matematyczno-Fizyczne 18 (1907), 1–59.