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

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