The representation of integers by positive quaternary quadratic forms

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 291-299

Résumé

Let $f(x,y,x,w)=x^2+y^2+z^2+D\omega^2$, where $D>1$ is an integer such that $D\ne d^2$ and $\sqrt{\mathstrut n}\big/\sqrt{\mathstrut D}=n^{\theta},0<\theta<1/2$. Let $r_f(n)$ be the number of representations of $n$ by $f$. It is proved that $$ r_f (n)=\pi^2\frac n{\sqrt D}\sigma_f(n)+O\biggl(\frac{n^{1+\varepsilon-c(\theta)}}{\sqrt D}\biggr), $$ where $\sigma_f(n)$ is the singular series, $c(\theta)>0$, and $\varepsilon$ is an arbitrarily small positive constant.