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

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler’s constant and $$\mathrm{\Delta }(x):=\sum _{n\le x}d(n)-x(\log x+2\gamma -1)$$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta (s)$ denote the Riemann zeta-function. It is shown that $${\int }_0^T\mathrm{\Delta }(t)\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^{2}dt\ll T(\log T{)}^4\mathrm{.}$$ Further, if $2\le k\le 8$isafixedinteger,thenweprovetheasymptoticformula$$ \int_1^{T}\Delta^{k}(t)|\zeta(1/2+it)|^2,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-\eta_k+\varepsilon}), where $c_1(k)$ and $c_2(k)$ are explicit constants, and where\eta_2= 3/20, \eta_3= \eta_4=1/10,\ \eta_5=3/80,\ \eta_6=35/4742,\ \eta_7=17/6312,\ \eta_8=8/9433.$$Theresultsdependonthepowermomentsof$\Delta(t)$and$E(T)$, the classical error term in the asymptotic formula for the mean square of $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$.