On the average orders of the error term in the Dirichlet divisor problem

Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas ∫1xΔ(u)kdu and ∑n⩽xΔ(n)k with a natural number k. In particular we study the case k=2 and 3 in detail.