Algebra & Number Theory

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

Pär Kurlberg and Carl Pomerance

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For coprime integers g and n, let g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of g(n) as nx 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 g(p) as px ranges over primes.

Article information

Algebra Number Theory, Volume 7, Number 4 (2013), 981-999.

Received: 25 August 2011
Revised: 3 March 2012
Accepted: 24 May 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N37: Asymptotic results on arithmetic functions

average multiplicative order


Kurlberg, Pär; Pomerance, Carl. On a problem of Arnold: The average multiplicative order of a given integer. Algebra Number Theory 7 (2013), no. 4, 981--999. doi:10.2140/ant.2013.7.981.

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