An average number of square-free values of polynomials

TL;DR

This paper investigates the average error term in counting square-free values of polynomials, showing under the Riemann hypothesis that it aligns with the expected order for polynomials of any degree.

Contribution

It proves, assuming the Riemann hypothesis, that the average error term for counting square-free polynomial values matches the conjectured order for all degrees.

Findings

Error term is $O_\varepsilon(N^{1/4+\varepsilon})$ on average

Result holds for polynomials of arbitrary degree

Assumption of Riemann hypothesis is crucial

Abstract

The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Usually, it is more difficult to obtain a similar order of error term for a higher degree polynomial $f(x)$ in place of $\text{Id}(x)$. Under the Riemann hypothesis, we show that the error term, on average in a weak sense, over polynomials of arbitrary degree, is of the expected order $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$.