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

Abstract

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.