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.
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.
@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://www.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, Tome 9 (1997) no. 1, pp. 25-39. http://www.numdam.org/item/JTNB_1997__9_1_25_0/
[Ch1] Some applications of large sieve in Riemann surfaces, Acta Arithmetica 77 (1996), 315-337. | MR | Zbl
,[Ch2] Correlated sums of r(n), Preprint, 1996. | MR
,[Fr] Einführung in die Gitterpunktlehre, vol. 73, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften; Math. Reihe, Birkhaüser Verlag, 1982. | MR | Zbl
,[Ga] De nexu inter multitudinem classium in quas formae binariae secundi gradus distribuntur, eaurumque determinatem [II], Werke V. 2, (1839), 269-291.
,[Go] Approximative Funktionalgleichungen und Mittelwertsätze Dirichletreihen, die Spitzenformen assoziiert sind, Comm. Math. Hel. 50 (1975), 327-361. | MR | Zbl
,[Gr-Ry] Tables of Integrals, Series and Products, Fifth edition, (Editor: A. Jeffrey), Academic Press, 1994. | MR | Zbl
and ,[Ha] The average order of the arithmetical functions P(x) and Δ(x), Proc. London Math. Soc. (2) 15 (1916), 192-213. | JFM
,[Hu] Exponential sums and lattice points II, Proc. London Math. Soc. (3) 66 (1993), 279-301. | MR | Zbl
,[Iv] Lectures on Mean Values of the Riemann Zeta-Function, Lectures on Mathematics and Physics, vol. 82, Tata institute of fundamental research, Springer Verlag, 1991. | MR | Zbl
,[Iw1] The spectral growth of automorphic L-functions, J. Reine Angew. Math. 428 (1992), 139-159. | MR | Zbl
,[Iw2] Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1995. | MR | Zbl
,[Iw-Sa] L∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. 141 (1995), 301-320. | Zbl
and ,[Kl] On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Mathematica 49 (1926), 407-464. | JFM
,[La] Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Zeitschrift 21 (1924), 126-132. | JFM | MR
,[Lu] The spectral mean value for linear forms in twisted coefficients of cusp forms, Acta Arithmetica 70 (1995), 377-391. | MR | Zbl
,[Ph-Ru] The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), 78-116. | MR | Zbl
and ,[Ra] Zum Ellipsoidproblem in algebraischen Zahlkörpen, Acta Arithmetica 58 (1991), 309-333. | MR | Zbl
,[Sc] Übertragung des Kreisproblems auf reell-quadratische Zahlkörper, Math. Ann. 145 (1962), 273-284. | MR | Zbl
,