On the average number of square-free values of polynomials

Igor E. Shparlinski

doi:10.4153/CMB-2012-021-8

On the average number of square-free values of polynomials

Igor E. Shparlinski^*

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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 ≥ N^k-1+ε for some fixed ε > 0. Individual results of this kind for polynomials of degree k > 3, due to A. Granville (1998), are only known under the ABC-conjecture.

Original language	English
Pages (from-to)	844-849
Number of pages	6
Journal	Canadian Mathematical Bulletin
Volume	56
Issue number	4
DOIs	https://doi.org/10.4153/CMB-2012-021-8
Publication status	Published - Dec 2013

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title = "On the average number of square-free values of polynomials",

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 ≥ Nk-1+ε for some fixed ε > 0. Individual results of this kind for polynomials of degree k > 3, due to A. Granville (1998), are only known under the ABC-conjecture.",

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AB - 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 ≥ Nk-1+ε for some fixed ε > 0. Individual results of this kind for polynomials of degree k > 3, due to A. Granville (1998), are only known under the ABC-conjecture.

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DO - 10.4153/CMB-2012-021-8

M3 - Article

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SP - 844

EP - 849

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

IS - 4

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On the average number of square-free values of polynomials

Abstract

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