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Joachim von zur Gathen; Arnold Knopfmacher; Florian Luca; Lutz G. Lucht; Igor E. Shparlinski
Average order in cyclic groups
Journal de théorie des nombres de Bordeaux, 16 no. 1 (2004), p. 107-123, doi: 10.5802/jtnb.436
Article PDF | Reviews MR 2145575 | Zbl 1079.11003 | 1 citation in Cedram

Résumé - Abstract

For each natural number $n$ we determine the average order $\alpha (n)$ of the elements in a cyclic group of order $n$. We show that more than half of the contribution to $\alpha (n)$ comes from the $\varphi (n)$ primitive elements of order $n$. It is therefore of interest to study also the function $\beta (n)=\alpha (n)/\varphi (n)$. We determine the mean behavior of $\alpha $, $\beta $, $1/\beta $, and also consider these functions in the multiplicative groups of finite fields.

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