On Sums of Two Coprime k-th Powers

• Autores: Wenguang Zhai
• Localización: Monatshefte für mathematik, ISSN 0026-9255, Vol. 144, Nº 3, 2005, págs. 233-250
• Idioma: inglés
• DOI: 10.1007/s00605-004-0293-9
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• Resumen

  • For fixed k3, let It is known that the asymptotic formula 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 (s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then Ek(x)=O(x4/15+), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of Ek(x) for k6 is also obtained, which implies that Ek(x)=(x1/k¿1/k2) for k6 under RH.