On certain integrals involving the Dirichlet divisor problem

We prove that $${\int }_1^X\mathrm{\Delta }(x){\mathrm{\Delta }}_3(x)dx\ll X^{13\mathrm{/}9}{\log }^{10\mathrm{/}3}X,{\int }_1^X\mathrm{\Delta }(x){\mathrm{\Delta }}_4(x)dx{\ll }_{\epsilon }{X}^{25\mathrm{/}16+\epsilon },$$ where ${\mathrm{\Delta }}_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by ${\zeta }^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for ${\mathrm{\Delta }}_k(x)$. We also obtain the analogues of the above bounds when $\text{D}(x)$ is replaced by $E(x)$, the error term in the mean square formula for $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$.