Density of positive diagonal binary quadratic forms
Volume 193 / 2020
Acta Arithmetica 193 (2020), 1-48
MSC: 11E16, 11P55, 11N56.
DOI: 10.4064/aa180519-8-4
Published online: 11 December 2019
Abstract
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 $z\ll (\log n)\log \log n$.
