Abstract: | For fixed k ≥ 3, let Ek(x) denote the error term of the sum
?n £ xrk(n)\sum_{n\le x}\rho_k(n)
, where
rk(n) = ?n=|m|k+|l|k, g.c.d.(m,l)=1\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
E3(x) << x331/1254+eE_3(x)\ll x^{331/1254+\varepsilon}
,
E4(x) << x37/184+eE_4(x)\ll x^{37/184+\varepsilon}
. A short interval result is also obtained. |