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 $d \leqslant Δ$, with $x,y $ integers? Writing $Δ$ as $(\log N)^{\log 2} 2^{α\sqrt{\log \log N}}$ for some $α\in (-\infty, \infty)$, we show that the answer is $Φ(α) + o(1)$, where $Φ$ is the Gaussian distribution function $Φ(α) = \frac{1}{2π} \int^α_{-\infty} e^{-x^2/2} dx$. 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 $d \leqslant (\log N)^{\log 2 + \varepsilon}$.