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Average order in cyclic groups

Joachim von zur GathenArnold KnopfmacherFlorian LucaLutz G. LuchtIgor E. Shparlinski — 2004

Journal de Théorie des Nombres de Bordeaux

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.

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