IRRATIONAL FACTOR OF ORDER k AND ITS CONNECTIONS WITH k-FREE INTEGERS

摘要

We introduce an irrational factor of order k defined by I_k(n) = ∏_i~l=1~p_i~(βi), where n = ∏_i~l=1~p_i~(αi) is the factorization of n and β_i = It turns out that the function I_k(n) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for ∏_(v=1)~n I_k(v)~(1), ∑_(n≤x) I_k(n) and ∑_(n≤x) (1 - n/x) I_k(n).