On the distribution of integers coprime to an integer free of small prime factors

摘要

Let M-k (q;h) be the kth moment of the number of integers coprime to q in an interval of length h centered on its mean h (q)/q. By comparison with the kth centered moment of the binomial distribution with parameters h and P, for which we show mu(k)(h, P) (ck)(k/2)(k + hP(1 - P))(k/2), uniformly in k, h and P and where c > 0 is an absolute constant, we prove the following upper bound M-k (q; h) (c'k)(k/2)q (k + h phi(q)/q)(k/2), where c' > 0 is an absolute constant, uniformly in k, h and q provided that every prime factor of q is greater than or equal to h. (C) 2008 Elsevier Inc. Tous droits reserves.