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) := Sigma(n <= 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 integral(T)(0) Delta(k)(t) = vertical bar zeta(1/2 + it)vertical bar(2m) dt (k, m is an element of N) are given. This complements the results of [A. Ivi ' c and W. Zhai, On some mean value results for vertical bar zeta(1/2 + it)vertical bar and a divisor problem II, Indag. Math. 26(5) (2015) 842-866], where asymptotic formulas for 2 <= k <= 8, m = 1 were established for the above integral.