On some upper bounds for the zeta-function and the Dirichlet divisor problem

Let $d(n)$ be the number of divisors of $n$, let $$\Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1)$$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$\int_0^T\Delta^k(t)|\zeta(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N)$$ are given. This complements the results of the paper Ivi\'c-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.