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

Let $Δ(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $Δ^*(x) = -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$. We show that $\int_T^{T+H}Δ^*\bigl(\frac{t}{2π}\bigr)|ζ(1/2+it)|^2dt \;\ll\; HT^{1/6}\log^{7/2}T \quad(T^{2/3+\varepsilon} \le H = H(T) \le T),$ $\int_0^TΔ(t)|ζ(1/2+it)|^2dt \;\ll\; T^{9/8}(\log T)^{5/2},$ and obtain asymptotic formulae for $\int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^2 |ζ(1/2+it)|^2dt,\quad \int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^3|ζ(1/2+it)|^2dt.$ The importance of the $Δ^*$-function comes from the fact that it is the analogue of $E(T)$, the error term in the mean square formula for $|ζ(1/2+it)|^2$. We also show, if $E^*(T) := E(T) - 2πΔ^*(T/(2π))$, $\int_0^T E^*(t)E^j(t)|ζ(1/2+it)|^2dt \; \ll_{j,\varepsilon}\; T^{7/6+j/4+\varepsilon}\quad(j= 1,2,3).$

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