Vol. 9, No. 1, 2016

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On the distribution of the greatest common divisor of Gaussian integers

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

Vol. 9 (2016), No. 1, 27–40
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.

Keywords
Gaussian integer, gcd, moment, Dedekind zeta function
Mathematical Subject Classification 2010
Primary: 11N37, 11A05, 11K65, 60E05
Milestones
Received: 27 March 2013
Revised: 9 January 2015
Accepted: 28 January 2015
Published: 17 December 2015

Communicated by Kenneth S. Berenhaut
Authors
 Tai-Danae Bradley Department of Mathematics The Graduate Center, CUNY 365 5th Avenue New York, NY 10016 United States Yin Choi Cheng Department of Mathematics The Graduate Center, CUNY 365 5th Avenue New York, NY 10016 United States Yan Fei Luo GACE Consulting Engineers PC 105 Madison Avenue 6th Floor New York, NY 10016 United States