Algebra & Number Theory

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

Abstract

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

Article information

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

Dates
Revised: 3 March 2012
Accepted: 24 May 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729988

Digital Object Identifier
doi:10.2140/ant.2013.7.981

Mathematical Reviews number (MathSciNet)
MR3095233

Zentralblatt MATH identifier
1282.11131

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions

Keywords
average multiplicative order

Citation

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

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