Sums of squares in $\mathbb{Z}\left[\sqrt{k}\right]$

Chamizo, Fernando

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 25-39.

Nous étudions une généralisation du fameux problème du cercle aux anneaux d'entiers quadratiques réels : nous nous intéressons à $C(N,M)={\sum }_{n\le N}{\sum }_{m\le M}r(n+m\sqrt{k})$, le nombre de représentations de $n+m\sqrt{k}$ comme somme de deux carrés dans $\mathbb{Z}\left[\sqrt{k}\right]$ (où $k>1$ et sans facteur carré). En utilisant la théorie spectrale dans $PS{L}_2\left(\mathbb{Z}\right)\setminus ℍ$, nous obtenons une formule asymptotique avec terme erreur pour $C(N,M)$, démontrant que certaines techniques d'estimations de fonctions $L$ automorphes fournissent précisément des majorations de ce terme erreur.

We study a generalization of the classical circle problem to real quadratic rings. Namely we study $C(N,M)={\sum }_{n\le N}{\sum }_{m\le M}r(n+m\sqrt{k})$ where $r(n+m\sqrt{k})$ is the number of representations of $n+m\sqrt{k}$ as a sum of two squares in $\mathbb{Z}\left[\sqrt{k}\right]$ (with $k>1$ and squarefree). Using spectral theory in $PS{L}_2\left(\mathbb{Z}\right)\setminus ℍ$, we get an asymptotic formula with error term for $C(N,M)$, showing that some techniques on the estimation of automorphic $L$-functions can be applied to get upper bounds of the error term.

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