On the Riemann zeta-function and the divisor problem III

Abstract

Let $\Delta (x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta (1/2+it)|$. If $E^{\ast }(t)=E(t)-2\pi {\Delta }^{\ast }(t/2\pi )$ with ${\Delta }^{\ast }(x)=-\Delta (x)+2\Delta (2x)-\frac{1}{2}\Delta (4x)$ and we set ${\int }_0^T{E}^{\ast }(t)dt=3\pi T/4+R(T)$, then we obtain $$R(T)=O_{\epsilon }(T^{593/912+\epsilon }),{\int }_0^T{R}^4(t)dt{\ll }_{\epsilon }{T}^{3+\epsilon },$$ and $${\int }_0^T{R}^2(t)dt=T^2{P}_3(\log T)+O_{\epsilon }(T^{11/6+\epsilon }),$$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.

18 pages

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