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Density of positive diagonal binary quadratic forms

Volume 193 / 2020

Brandon Hanson, Robert Vaughan Acta Arithmetica 193 (2020), 1-48 MSC: 11E16, 11P55, 11N56. DOI: 10.4064/aa180519-8-4 Published online: 11 December 2019


We are concerned with those $n$ which can be represented in the form $$x^2+zy^2=n$$ where $z$ is generally small compared with the size of $n$. We show that almost all $n$ have such a representation with $$z\le (\log n) (\log \log n)^{3+\delta }$$ and that a positive proportion do with $z\ll (\log n)\log \log n$.


  • Brandon HansonDepartment of Mathematics
    Boyd Graduate Studies Research Center
    University of Georgia
    Athens, GA 30602, U.S.A.
  • Robert VaughanDepartment of Mathematics
    Pennsylvania State University
    University Park, PA 16802, U.S.A.

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