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 $γ$ denote Euler's constant and $Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1)$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. It is shown that $\int_0^TΔ(t)|ζ(1/2+it)|^2\,dt \ll T(\log T)^{4}.$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $\int_1^{T}Δ^{k}(t)|ζ(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-η_k+\varepsilon}),$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and where $η_2= 3/20, η_3= η_4=1/10,\ η_5=3/80,\ η_6=35/4742,\ η_7=17/6312,\ η_8=8/9433.$ The results depend on the power moments of $Δ(t)$ and $E(T)$, the classical error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$.

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