Average multiplicative orders of elements modulo n

Arithmetic functions with linear recurrence sequences

“…Clearly, these two functions are multiplicative and this has been shown both in [5] as well as in [9] in a wider context. We notice that a formula for α(n)/n when n = p α is a prime power appears in [5] and also, in a different form in [9], which is more suitable for our purposes. In particular, Lemma 3 from [9] gives…”

Artin's Primitive Root Conjecture – A Survey

“…Note that u(n) ≤ λ(n). They proved various results comparing u(n) with λ(n), see also [312,313,343].…”

Some Mean Values Related to Average Multiplicative Orders of Elements in Finite Fields

“…The first direction consists in replacing the cyclic group Z n by any finite abelian group and investigating the analogs of the functions α(n), α(n)/n, and β(n) in this more general setting. For a finite abelian group G, the function α(G) can be taken to be the average value of the orders of elements in G, the analog of the function α(n)/n can be taken to be α(G)/e(G), where e(G) denotes the maximal order of elements in G, and finally the analog of the function β(n) can be taken to be the function (α(G)/e(G))/(γ (G)/#G), where γ (G) denotes the number of elements of maximal order e(G) in the group G. These functions were investigated in [5] and most of the properties of these functions proved in [8] have been shown to hold in this wider context. The paper [5] contains also some applications of this general formalism to the special case when G := U (Z n ) is the group of invertible elements modulo n.…”