On the higher moments of the error term in the divisor problem

Abstract

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X Δ^3(x){\rm d}x = BX^{7/4} + O_ε(X^{β+ε}) \qquad(B > 0) $$ and $$ \int_1^X Δ^4(x){\rm d}x = CX^2 + O_ε(X^{γ+ε}) \qquad(C > 0) $$ with $β= 7/5, γ= 23/12$. This improves on the values $β= 47/28, γ= 45/23$, due to K.-M. Tsang. A result on the integrals of $Δ^3(x)$ and $Δ^4(x)$ in short intervals is also proved.