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

We study a generalization of the classical circle problem to real quadratic rings. Namely we study C(N,M)= nN mM r(n+mk) where r ( n + m k ) is the number of representations of n + m k as a sum of two squares in [k] (with k>1 and squarefree). Using spectral theory in PSL 2 (), 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)= nN mM r(n+mk), le nombre de représentations de n + m k comme somme de deux carrés dans [k] (où k>1 et sans facteur carré). En utilisant la théorie spectrale dans PSL 2 (), 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.

@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},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     pages = {25-39},
     zbl = {0924.11081},
     mrnumber = {1469659},
     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, Volume 9 (1997) no. 1, pp. 25-39. http://www.numdam.org/item/JTNB_1997__9_1_25_0/

[Ch1] F. Chamizo, Some applications of large sieve in Riemann surfaces, Acta Arithmetica 77 (1996), 315-337. | MR 1414513 | Zbl 0863.11062

[Ch2] F. Chamizo, Correlated sums of r(n), Preprint, 1996. | MR 1661040

[Fr] F. Fricker, Einführung in die Gitterpunktlehre, vol. 73, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften; Math. Reihe, Birkhaüser Verlag, 1982. | MR 673938 | Zbl 0489.10001

[Ga] C.F. Gauss, De nexu inter multitudinem classium in quas formae binariae secundi gradus distribuntur, eaurumque determinatem [II], Werke V. 2, (1839), 269-291.

[Go] A. Good, Approximative Funktionalgleichungen und Mittelwertsätze Dirichletreihen, die Spitzenformen assoziiert sind, Comm. Math. Hel. 50 (1975), 327-361. | MR 401651 | Zbl 0315.10038

[Gr-Ry] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Fifth edition, (Editor: A. Jeffrey), Academic Press, 1994. | MR 1243179 | Zbl 0918.65002

[Ha] G.H. Hardy, The average order of the arithmetical functions P(x) and Δ(x), Proc. London Math. Soc. (2) 15 (1916), 192-213. | JFM 46.0262.01

[Hu] M.N. Huxley, Exponential sums and lattice points II, Proc. London Math. Soc. (3) 66 (1993), 279-301. | MR 1199067 | Zbl 0820.11060

[Iv] A. Ivić, 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 1230387 | Zbl 0758.11036

[Iw1] H. Iwaniec, The spectral growth of automorphic L-functions, J. Reine Angew. Math. 428 (1992), 139-159. | MR 1166510 | Zbl 0746.11024

[Iw2] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1995. | MR 1325466 | Zbl 0847.11028

[Iw-Sa] H. Iwaniec and P. Sarnak, L∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. 141 (1995), 301-320. | Zbl 0833.11019

[Kl] H.D. Kloosterman, On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Mathematica 49 (1926), 407-464. | JFM 53.0155.01

[La] E. Landau, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Zeitschrift 21 (1924), 126-132. | JFM 50.0118.01 | MR 1544690

[Lu] W. Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms, Acta Arithmetica 70 (1995), 377-391. | MR 1330741 | Zbl 0821.11035

[Ph-Ru] R.S. Phillips and Z. Rudnick, The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), 78-116. | MR 1270589 | Zbl 0812.11035

[Ra] U. Rausch, Zum Ellipsoidproblem in algebraischen Zahlkörpen, Acta Arithmetica 58 (1991), 309-333. | MR 1121090 | Zbl 0734.11050

[Sc] W. Schaal, Übertragung des Kreisproblems auf reell-quadratische Zahlkörper, Math. Ann. 145 (1962), 273-284. | MR 142539 | Zbl 0099.03603