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

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.

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

Communicated by Kenneth S. Berenhaut
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