Vol. 7, No. 4, 2013

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On a problem of Arnold: The average multiplicative order of a given integer

Pär Kurlberg and Carl Pomerance

Vol. 7 (2013), No. 4, 981–999
Abstract

For coprime integers $g$ and $n$, let ${\ell }_{g}\left(n\right)$ denote the multiplicative order of $g$ modulo $n$. Motivated by a conjecture of Arnold, we study the average of ${\ell }_{g}\left(n\right)$ as $n\le x$ ranges over integers coprime to $g$, and $x$ tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of ${\ell }_{g}\left(p\right)$ as $p\le x$ ranges over primes.

Keywords
average multiplicative order
Primary: 11N37
Milestones
Received: 25 August 2011
Revised: 3 March 2012
Accepted: 24 May 2012
Published: 29 August 2013
Authors
 Pär Kurlberg Department of Mathematics KTH Royal Institute of Technology SE-100 44 Stockholm Sweden http://www.math.kth.se/~kurlberg/ Carl Pomerance Mathematics Department Kemeny Hall Dartmouth College Hanover NH 03755 United States http://www.math.dartmouth.edu/~carlp