Average order in cyclic groups

Abstract

Résumé. Pour chaque entier naturel n, nous déterminons l'ordre moyen α(n) deséléments du groupe cyclique d'ordre n. Nous montrons que plus de la moitié de la contributionà α(n) provient des ϕ(n)éléments primitifs d'ordre n. Il est par conséquent intéressant d'étudierégalement la fonction β(n) = α(n)/ϕ(n). Nous détermi-nons le comportement moyen de α, β, 1/β et considérons aussi ces fonctions dans le cas du groupe multiplicatif d'un corps fini.Abstract. For each natural number n we determine the average order α(n) of the elements in a cyclic group of order n. We show that more than half of the contribution to α(n) comes from the ϕ(n) primitive elements of order n. It is therefore of interest to study also the function β(n) = α(n)/ϕ(n). We determine the mean behavior of α, β, 1/β, and also consider these functions in the multiplicative groups of finite fields.

Average multiplicative orders of elements modulo n

“…In [8], the function α(n) which gives the average additive order of elements modulo n has been considered, and several of its properties have been investigated, such as mean value, minimal and maximal order, and so on. The behaviour of this function restricted only to shifted primes (say, only to positive integers n of the form p − 1 with p a prime number), or to numbers of the form 2 n − 1 has been investigated in [8] and [13].…”

Artin's Primitive Root Conjecture – A Survey

“…30) Average multiplicative order. J. von zur Gathen et al [172] defined u(n) to be the average multiplicative order of the elements of (Z/nZ) * . Note that u(n) ≤ λ(n).…”

Computing special powers in finite fields

“…Thus we can compute an eth power of an element in F × 2 22 (or in any ring) with 2 (11) = 5 multiplications, using the first addition chain of Example 1 restricted to 11, plus (11 − 1) · 2 = 20 squarings.…”

“…Thus β is primitive if and only if β (q n −1)/p = 1 for all primes p dividing q n − 1. See [22] for the average order in F × q n and [7] for computing large primitive trinomials over F 2 .…”

“…Thus, we can test an element in F × 2 22 for primitivity using 5 + 2 · 1 + 1 + 9 = 17 multiplications and 10 · 2 + 1 · 11 + 1 · 11 + 0 + 12 = 54 squarings in F 2 22 .…”