On Sums of Two Coprime k-th Powers

For fixed k3, let It is known that the asymptotic formula holds for some constant c_k. Let E_k(x)=R_k(x)–c_kx^2/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 (s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then E_k(x)=O(x^4/15+), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of E_k(x) for k6 is also obtained, which implies that E_k(x)=(x^1/k–1/k2) for k6 under RH.