Duke Mathematical Journal

Estimates for representation numbers of quadratic forms

Valentin Blomer and Andrew Granville

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Let f be a primitive positive integral binary quadratic form of discriminant D, and let rf(n) be the number of representations of n by f up to automorphisms of f. In this article, we give estimates and asymptotics for the quantity nxrf(n)β for all β0 and uniformly in D=o(x). As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers

Article information

Duke Math. J., Volume 135, Number 2 (2006), 261-302.

First available in Project Euclid: 17 October 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E16: General binary quadratic forms
Secondary: 11N56: Rate of growth of arithmetic functions


Blomer, Valentin; Granville, Andrew. Estimates for representation numbers of quadratic forms. Duke Math. J. 135 (2006), no. 2, 261--302. doi:10.1215/S0012-7094-06-13522-6. https://projecteuclid.org/euclid.dmj/1161093265

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