An average number of square-free values of polynomials

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 $O_{\epsilon }(N^{\frac{1}{4}+\epsilon })$ for arbitrarily small $\epsilon >0$. 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 $O_{\epsilon }(N^{\frac{1}{4}+\epsilon })$.

7 pages