Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T19:36:01.108Z Has data issue: false hasContentIssue false
  • English
  • Français

POWER-FREE VALUES OF THE POLYNOMIAL t1⋯tr−1

Part of: Multiplicative number theory

Published online by Cambridge University Press:  23 September 2011

PIERRE LE BOUDEC
PIERRE LE BOUDEC*
Affiliation:
Université Denis Diderot (Paris VII), Institut de Mathématiques de Jussieu, UMR 7586, Case 7012 - Bâtiment Chevaleret, Bureau 7C14, 75205 Paris Cedex 13, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k,r≥2 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t1⋯tr−1 over [1,x]r∩ℤr.

Keywords

power-free values of polynomialsequidistribution in arithmetic progressionsKloosterman sums

MSC classification

Secondary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 154 - 163
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[Apo70]Apostol, T. M., ‘Möbius functions of order k’, Pacific J. Math. 32 (1970), 21–27.CrossRefGoogle Scholar
[Bro11]Browning, T. D., ‘Power-free values of polynomials’, Arch. Math. (Basel) 96(2) (2011), 139–150.CrossRefGoogle Scholar
[Bur63]Burgess, D. A., ‘On character sums and L-series. II’, Proc. Lond. Math. Soc. (3) 13 (1963), 524–536.CrossRefGoogle Scholar
[Bur86]Burgess, D. A., ‘The character sum estimate with r=3’, J. Lond. Math. Soc. (2) 33 (1986), 219–226.CrossRefGoogle Scholar
[Del74]Deligne, P., ‘La conjecture de Weil. I’, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.CrossRefGoogle Scholar
[FI85]Friedlander, J. B. and Iwaniec, H., ‘Incomplete Kloosterman sums and a divisor problem’, Ann. of Math. (2) 121(2) (1985), 319–350, with an appendix by B. J. Birch and E. Bombieri.CrossRefGoogle Scholar
[Gra98]Granville, A., ‘ ABC allows us to count squarefrees’, Int. Math. Res. Not. IMRN (19) (1998), 991–1009.CrossRefGoogle Scholar
[HB86]Heath-Brown, D. R., ‘The divisor function d 3(n) in arithmetic progressions’, Acta Arith. 47(1) (1986), 29–56.CrossRefGoogle Scholar
[Poo03]Poonen, B., ‘Squarefree values of multivariable polynomials’, Duke Math. J. 118(2) (2003), 353–373.CrossRefGoogle Scholar
[Shp07]Shparlinski, I. E., ‘On the distribution of points on multidimensional modular hyperbolas’, Proc. Japan Acad. Ser. A Math. Sci. 83(2) (2007), 5–9.CrossRefGoogle Scholar
[Smi79]Smith, R. A., ‘On n-dimensional Kloosterman sums’, J. Number Theory 11(3) (1979), 324–343, S. Chowla Anniversary Issue.CrossRefGoogle Scholar
[Tol10]Tolev, D., ‘On the number of pairs of positive integers x,y≤H such that x 2+y 2+1 is squarefree’, Monatsh. Math. (2010) (submitted) arXiv:1007.0353v2.CrossRefGoogle Scholar
[Wei48]Weil, A., Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Scientifiques et Industrielles, 1041 (Hermann, Paris, 1948).Google Scholar
[Wei81]Weinstein, L., ‘The hyper-Kloosterman sum’, Enseign. Math. (2) 27(1–2) (1981), 29–40.Google Scholar