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 $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. It is shown that $$ \int_0^T\Delta(t)|\zeta(1/2+it)|^2\,dt \ll T(\log T)^{4}. $$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \int_1^{T}\Delta^{k}(t)|\zeta(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-\eta_k+\varepsilon}), $$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and where $$\eta_2= 3/20, \eta_3= \eta_4=1/10,\ \eta_5=3/80,\ \eta_6=35/4742,\ \eta_7=17/6312,\ \eta_8=8/9433.$$ The results depend on the power moments of $\Delta(t)$ and $E(T)$, the classical error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$.