On the mean square of the zeta-function and the divisor problem

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 ${\Delta }^{\ast }(x)=-\Delta (x)+2\Delta (2x)-\frac{1}{2}\Delta (4x)$, then we obtain the asymptotic formula $${\int }_0^T(E^{\ast }(t){)}^2\mathrm{d}t=T^{4/3}{P}_3(\log T)+O_{\epsilon }(T^{7/6+\epsilon }),$$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.

10 pages

Cited by in corpus (1)

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