We study a generalization of the classical circle problem to real quadratic rings. Namely we study where is the number of representations of as a sum of two squares in (with and squarefree). Using spectral theory in , we get an asymptotic formula with error term for , showing that some techniques on the estimation of automorphic -functions can be applied to get upper bounds of the error term.
Nous étudions une généralisation du fameux problème du cercle aux anneaux d'entiers quadratiques réels : nous nous intéressons à , le nombre de représentations de comme somme de deux carrés dans (où et sans facteur carré). En utilisant la théorie spectrale dans , nous obtenons une formule asymptotique avec terme erreur pour , démontrant que certaines techniques d'estimations de fonctions automorphes fournissent précisément des majorations de ce terme erreur.
@article{JTNB_1997__9_1_25_0, author = {Chamizo, Fernando}, title = {Sums of squares in $\mathbb {Z}[\sqrt{k}]$}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {25--39}, publisher = {Universit\'e Bordeaux I}, volume = {9}, number = {1}, year = {1997}, mrnumber = {1469659}, zbl = {0924.11081}, language = {en}, url = {http://archive.numdam.org/item/JTNB_1997__9_1_25_0/} }
Chamizo, Fernando. Sums of squares in $\mathbb {Z}[\sqrt{k}]$. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 25-39. http://archive.numdam.org/item/JTNB_1997__9_1_25_0/
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