The distribution of k-free numbers

Let k ∈ {3; 4; 5}. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_k (x) = \sum\limits_{n \leqq xnisk - free} {1 - \frac{x} {{\zeta (k)}}} . $$\end{document} We give new upper bounds for Rk(x) conditional on the Riemann hypothesis, improving work of S. W. Graham and J. Pintz. The method stays close to that devised by H. L. Montgomery and R. C. Vaughan, with the improvement depending on exponential sum results.