Geodesic
On the Average Number of Square-Free Values of Polynomials
Canadian mathematical bulletin, Tome 56 (2013) no. 4, pp. 844-849

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DOI

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 $\varepsilon \,>\,0$ . Individual results of this kind for polynomials of degree $k\,>\,3$ , due to A. Granville (1998), are only known under the $ABC$ -conjecture.
DOI : 10.4153/CMB-2012-021-8
Mots-clés : 11N32, polynomials, square-free numbers 
Shparlinski, Igor E. On the Average Number of Square-Free Values of Polynomials. Canadian mathematical bulletin, Tome 56 (2013) no. 4, pp. 844-849. doi: 10.4153/CMB-2012-021-8
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     author = {Shparlinski, Igor E.},
     title = {On the {Average} {Number} of {Square-Free} {Values} of {Polynomials}},
     journal = {Canadian mathematical bulletin},
     pages = {844--849},
     year = {2013},
     volume = {56},
     number = {4},
     doi = {10.4153/CMB-2012-021-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2012-021-8/}
}
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