Covering integers by $x^2+d{y}^2$

Abstract

What proportion of integers $n⩽N$ may be expressed as $x^2+d{y}^2$ for some $d⩽\hat{I}”$, with $x,y$ integers? Writing $\hat{I}”$ as $(\log N{)}^{\log 2}{2}^{\hat{I}\pm \sqrt{\log \log N}}$ for some $\hat{I}\pm \in (-\infty ,\infty )$, we show that the answer is $\hat{I}¦(\hat{I}\pm )+o(1)$, where $\hat{I}¦$ is the Gaussian distribution function $\hat{I}¦(\hat{I}\pm )=\frac{1}{2\ddot{I}€}{\int }^{\hat{I}}{\pm }_{-\infty }{e}^{-x^2/2}dx$. A consequence of this is a phase transition: almost none of the integers $n⩽N$ can be represented by $x^2+d{y}^2$ with $d⩽(\log N{)}^{\log 2-\epsilon }$, but almost all of them can be represented by $x^2+d{y}^2$ with $d⩽(\log N{)}^{\log 2+\epsilon }$.

35 pages, to appear in Journal of the Institute of Mathematics of Jussieu. Minor revisions from first version