Abstract
It is not difficult to find an asymptotic formula for the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree. In the present paper we improve the estimate for the error term in this formula using the properties of certain exponential sums. A.Weils’s estimate for the Kloosterman sum plays the major role in our analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Estermann T.: Einige Sätze über quadratfreie Zahlen. Math. Ann. 105, 653–662 (1931)
Estermann T.: A new application of the Hardy–Littlewood–Kloosterman method. Proc. Lond. Math. Soc. 12, 425–444 (1962)
Filaseta M.: Powerfree values of binary forms. J. Number Theory 49(2), 250–268 (1994)
Greaves G.: Power-free values of binary forms. Q. J. Math. 43, 45–65 (1992)
Heath-Brown D.R.: The square sieve and consecutive square-free numbers. Math. Ann. 266, 251–259 (1984)
Hooley C.: Applications of sieve methods to the theory of numbers. Cambridge University Press, Cambridge (1976)
Hooley, C.: On the power-free values of polynomials in two variables, Roth 80th birthday volume (in press)
Hooley C.: On the power-free values of polynomials in two variables: II. J. Number Theory 129, 1443–1455 (2009)
Hua L.-K.: Introduction to number theory. Springer, Berlin (1982)
Iwaniec, H., Kowalski, E.: Analytic number theory. Colloquium Publications, vol. 53, Am. Math. Soc., (2004)
Poonen B.: Squarefree values of multivariable polynomials. Duke Math. J. 118(2), 353–373 (2003)
Tolev D.: On the exponential sum with squarefree numbers. Bull. Lond. Math. Soc. 37(6), 827–834 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Johannes Schoißengeier.
Rights and permissions
About this article
Cite this article
Tolev, D.I. On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree. Monatsh Math 165, 557–567 (2012). https://doi.org/10.1007/s00605-010-0246-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-010-0246-4