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

Let $d(n)$ be the number of divisors of $n$, let $Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1)$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $\int_0^TΔ^k(t)|ζ(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N)$ are given. This complements the results of the paper Ivić-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.

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