Geodesic
The representation of integers by positive quaternary quadratic forms
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 291-299 
O. M. Fomenko. The representation of integers by positive quaternary quadratic forms. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 291-299. http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a13/
@article{ZNSL_2001_276_a13,
     author = {O. M. Fomenko},
     title = {The representation of integers by positive quaternary quadratic forms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {291--299},
     year = {2001},
     volume = {276},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a13/}
}
TY  - JOUR
AU  - O. M. Fomenko
TI  - The representation of integers by positive quaternary quadratic forms
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2001
SP  - 291
EP  - 299
VL  - 276
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a13/
LA  - ru
ID  - ZNSL_2001_276_a13
ER  - 
%0 Journal Article
%A O. M. Fomenko
%T The representation of integers by positive quaternary quadratic forms
%J Zapiski Nauchnykh Seminarov POMI
%D 2001
%P 291-299
%V 276
%U http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a13/
%G ru
%F ZNSL_2001_276_a13

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.