On the $k$-free divisor problem

Abstract

Let ${\Delta }^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\ge 2$. In this paper we establish an asymptotic formula of the integral ${\int }_1^T|{\Delta }^{(k)}(x){|}^{2}dx$ for each $k\ge 4.$