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.
Pär Kurlberg. Carl Pomerance. "On a problem of Arnold: The average multiplicative order of a given integer." Algebra Number Theory 7 (4) 981 - 999, 2013. https://doi.org/10.2140/ant.2013.7.981