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arXiv · 1108.5209

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

Par Kurlberg·Carl Pomerance
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Abstract

For g,n coprime integers, 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 <= 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 <= x ranges over primes.

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Par Kurlberg, Carl Pomerance. 2011-08-25. On a problem of Arnold: the average multiplicative order of a given integer. https://arxiv.org/abs/1108.5209

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