On Sums of Two Coprime k-th Powers

摘要

For fixed k≥3, let $rho_k(n)=sum_{n=vert mvert^k+vert lvert^k, {rm g.c.d}(m,l)=1}1.$ It is known that the asymptotic formula $R_k(x)=sum_{nle x}rho_k(n)=c_k x^{2/k}+O(x^{1/k})$ holds for some constant c k . Let E k (x)=R k (x)−c k x 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 ζ(s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then E k (x)=O(x 4/15+ɛ), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of E k (x) for k≥6 is also obtained, which implies that E k (x)=Ω(x 1/k−1/k2) for k≥6 under RH.