Square free values of the order function

摘要

Given a∈Z {±1,0}, we consider the problemof enumerating the integers m coprime to a such that the orderof a modulo m is square free. This question is raised inanalogy to a result recently proved jointly with F. Saidak and I.Shparlinski where square free values of the Carmichael functionare studied. The technique is the one of Hooley that uses theChebotarev Density Theorem to enumerate primes for which the indexip(a) of a modulo p is divisible by a given integer.