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

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

2011-08-25

arxiv.org/abs/1108.5209v1

Mathematics

Number Theory

18 pages

Scientific paper

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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