On the Number of Square-Free Values of Polynomials on Average

http://dx.doi.org/10.4153/CMB-2012-021-8
6 pages

  Published:2012-06-22
 

We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture.