On some mean value results for the zeta-function and a divisor problem

Abstract

Let $\Delta (x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ${\Delta }^{\ast }(x)=-\Delta (x)+2\Delta (2x)-\frac{1}{2}\Delta (4x)$. We show that $${\int }_T^{T+H}{\Delta }^{\ast }\left(\frac{t}{2\pi }\right)|\zeta (1/2+it){|}^{2}dt\ll H{T}^{1/6}{\log }^{7/2}T(T^{2/3+\epsilon }\le H=H(T)\le T),$$ $${\int }_0^T\Delta (t)|\zeta (1/2+it){|}^{2}dt\ll T^{9/8}(\log T{)}^{5/2},$$ and obtain asymptotic formulae for $${\int }_0^T{\left({\Delta }^{\ast }\right(\frac{t}{2\pi }\left)\right)}^2|\zeta (1/2+it){|}^{2}dt,{\int }_0^T{\left({\Delta }^{\ast }\right(\frac{t}{2\pi }\left)\right)}^3|\zeta (1/2+it){|}^{2}dt.$$ The importance of the ${\Delta }^{\ast }$-function comes from the fact that it is the analogue of $E(T)$, the error term in the mean square formula for $|\zeta (1/2+it){|}^2$. We also show, if $E^{\ast }(T):=E(T)-2\pi {\Delta }^{\ast }(T/(2\pi ))$, $${\int }_0^T{E}^{\ast }(t)E^j(t)|\zeta (1/2+it){|}^{2}dt{\ll }_{j,\epsilon }{T}^{7/6+j/4+\epsilon }(j=1,2,3).$$

16 pages