Sums of squares in ℤ[k]
Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 25-39

We study a generalization of the classical circle problem to real quadratic rings. Namely we study C(N,M)=∑ n≤N ∑ m≤M r(n+mk) where r ( n + m k ) is the number of representations of n + m k as a sum of two squares in ℤ[k] (with k>1 and squarefree). Using spectral theory in PSL 2 (ℤ)∖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.

Nous étudions une généralisation du fameux problème du cercle aux anneaux d'entiers quadratiques réels : nous nous intéressons à C(N,M)=∑ n≤N ∑ m≤M r(n+mk), le nombre de représentations de n + m k comme somme de deux carrés dans ℤ[k] (où k>1 et sans facteur carré). En utilisant la théorie spectrale dans PSL 2 (ℤ)∖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.

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Fernando Chamizo. Sums of squares in $\mathbb {Z}[\sqrt{k}]$. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 25-39. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_25_0/

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