On Sums of Two Coprime k-th Powers

For fixed k≥3, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_k(n)=\sum_{n=\vert m\vert^k+\vert l\vert^k, {\rm g.c.d}(m,l)=1}1.$$\end{document} It is known that the asymptotic formula \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k(x)=\sum_{n\le x}\rho_k(n)=c_k x^{2/k}+O(x^{1/k})$$\end{document} holds for some constant ck. Let Ek(x)=Rk(x)−ckx2/k. We cannot improve the exponent 1/k at present if we do not have further knowledge about the distribution of the zeros of the Riemann Zeta function ζ(s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then Ek(x)=O(x4/15+ɛ), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of Ek(x) for k≥6 is also obtained, which implies that Ek(x)=Ω(x1/k−1/k2) for k≥6 under RH.