  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.

Number Theory