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

For any positive integer n let α(n) denote the average order of elements in the cyclic group Z_n. In this note, we investigate the functions α(n)/n and α(n)/φ(n) when n ranges through numbers of the form p−1 with p prime, and when n ranges through numbers of the form 2^m−1 with m a positive integer. In particular, we show that such functions have limiting distributions, and we compute their average values, and their minimal and maximal orders.To Jean-Louis Nicolas at his 60th birthday2000 Mathematics Subject Classification: Primary—11N45; Secondary—05A16, 11N37This work was supported in part by Grant SEP-CONACYT 37259-E.