On the average number of square-free values of polynomials (Q2862964)

For a polynomial \(f(X)\in {\mathbb Z}[X]\) let \(S_f(N)\) be the number of \(n\leq N\) such that \(f(n)\) is square-free. Let \({\mathcal F}_k(H)\) be the set of all primitive polynomials with integer coefficients of degree at most \(k\) and naïve height at most \(H\). The main result of the paper is the estimate NEWLINE\[NEWLINE \frac{1}{\# {\mathcal F}_k(H)} \sum_{f\in {\mathcal F}_k(H)} |S_f(N)-c_f N|\leq N^{1/2} H^{o(1)}+N^{(k+1)/2} H^{-1/2+o(1)} NEWLINE\]NEWLINE as \(H\to \infty\), where NEWLINE\[NEWLINE c_f=\prod_{p~{\text{prime}}} \left(1-\frac{\rho_f(p^2)}{p^2}\right) NEWLINE\]NEWLINE and \(\rho_f(m)\) denotes the number of solutions of the congruence \(f(n)\equiv 0\pmod m\). As a consequence, the author obtains that for most polynomials \(f\in {\mathcal F}_k(H)\) the estimate \(S_f(N)\sim c_f N\) holds, provided that \(N^{A}\geq H\geq N^{k-1+\varepsilon}\) for some fixed \(A\) and \(\varepsilon>0\). Individual results of this type for polynomials of degree \(k>3\) are known only conditionally under the ABC conjecture by work of \textit{A. Granville} [Int. Math. Res. Not. 1998, No. 19, 991--1009 (1998; Zbl 0924.11018)].