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

Kurlberg, Pär; Pomerance, Carl

doi:10.2140/ant.2013.7.981

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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

DOI: 10.2140/ant.2013.7.981

Abstract

For coprime integers $g$ and $n$ , let $ℓ_{g} (n)$ denote the multiplicative order of $g$ modulo $n$ . Motivated by a conjecture of Arnold, we study the average of $ℓ_{g} (n)$ as $n \leq x$ 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 $ℓ_{g} (p)$ as $p \leq x$ ranges over primes.

Keywords

average multiplicative order

Mathematical Subject Classification 2010

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

Vol. 7, No. 4, 2013