Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 4 (2013), 981-999.
On a problem of Arnold: The average multiplicative order of a given integer
For coprime integers and , let denote the multiplicative order of modulo . Motivated by a conjecture of Arnold, we study the average of as ranges over integers coprime to , and 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 as ranges over primes.
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
Permanent link to this document
Digital Object Identifier
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. https://projecteuclid.org/euclid.ant/1513729988