Covering integers by $x^2 + dy^2$

Abstract

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