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3 results

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

  • Igor E. Shparlinski
  • Journal:
    Canadian Mathematical Bulletin / Volume 56 / Issue 4 / 01 December 2013
    Published online by Cambridge University Press:
    20 November 2018, pp. 844-849
    Print publication:
    01 December 2013
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  • 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.

  • Square-free values of quadratic polynomials

  • Part of
  • J. B. Friedlander, H. Iwaniec
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 53 / Issue 2 / June 2010
    Published online by Cambridge University Press:
    30 April 2010, pp. 385-392
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  • We consider the quadratic polynomial m2 + D and study the asymptotic formula for the number of integers m, 1≤ m ≤ M, for which the values of the polynomial are square-free. We are interested in particular in the question of how small we can take m in relation to D and still have the asymptotic hold.