On the asymptotic behavior of sums ∑n≤xf(n){x/n}k

摘要

Let f(t) be an arbitrary real-valued positive nondecreasing function, in this paper we prove that Sf,k(x) − Tf,k(x) = ?(f(x)x131/416(log x)26947/8320),Sf,k(x) − Tf,k(x) = Sf,1(x) − Tf,1(x) + ?(f(x)x227/796+?), where Sf,k(x) is the sum given in the title, Tf,k(x) =∫1xf(t){x t }kdt and k is a positive integer. This improves the result of Mercier and Nowak.

关键词