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

Published:2012-06-22
Printed: Dec 2013
• Igor E. Shparlinski,
Department of Computing, Macquarie University, NSW 2109, Australia
 Format: LaTeX MathJax PDF

## Abstract

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.
 Keywords: polynomials, square-free numbers
 MSC Classifications: 11N32 - Primes represented by polynomials; other multiplicative structure of polynomial values

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