On the Riemann zeta-function and the divisor problem II

Abstract

First part of this paper was published in CEJM (2)(4) (2004), 1-15. It is proved now that $${\int }_0^T|E^{\ast }(t){|}^5\mathrm{d}t{\ll }_{\epsilon }{T}^{2+\epsilon }.$$ Here $$E^{\ast }(t)=E(t)-2\pi {\Delta }^{\ast }(t/2\pi ),{\Delta }^{\ast }(x)=-\Delta (x)+2\Delta (2x)-\frac{1}{2}\Delta (4x),$$ where $E(t)$ is the error term in the mean square formula for $|\zeta (1/2+it)|$ and $\Delta (x)$ is the error term in the Dirichlet divisor problem. It is also shown how bounds for moments of $|E^{\ast }(t)|$ lead to bounds for moments of $|\zeta (1/2+it)|$.

13 pages