On Sums of Two Coprime k-th Powers(II)

摘要

For fixed k ≥ 3, let E k (x) denote the error term of the sum $sum_{nle x}rho_k(n)$ , where $rho_k(n) = sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1} $ 1. It is proved that if the Riemann hypothesis is true, then $E_3(x)ll x^{331/1254+varepsilon}$ , $E_4(x)ll x^{37/184+varepsilon}$ . A short interval result is also obtained.