On the $k$-free divisor problem

TL;DR

This paper derives an asymptotic formula for the mean square of the error term in the $k$-free divisor problem for all $k \\geq 4$, advancing understanding of divisor-related error estimates.

Contribution

It establishes an asymptotic formula for the integral of the squared error term in the $k$-free divisor problem for all $k \\geq 4$, filling a gap in divisor problem research.

Findings

Asymptotic formula derived for the mean square of the error term.

Results apply to all $k \\geq 4$, extending previous work.

Provides new insights into the distribution of $k$-free divisors.

Abstract

Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$