On some mean square estimates for the zeta-function in short intervals

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 ${\Delta }^{\ast }(x)=-\Delta (x)+2\Delta (2x)-1/2\Delta (4x)$ and we set ${\int }_0^T{E}^{\ast }(t)dt=3\pi T/4+R(T)$, then we obtain $${\int }_T^{T+H}(E^{\ast }(t){)}^{2}dt\gg H{T}^{1/3}{\log }^{3}T$$ and $$HT{\log }^{3}T\ll {\int }_T^{T+H}{R}^2(t)dt\ll HT{\log }^{3}T,$$ for $T^{2/3+\epsilon }\le H\le T$.

14 pages

References in corpus (1)

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