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Fernando Chamizo
Sums of squares in $\mathbb{Z}[\sqrt{k}]$
Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), p. 25-39, doi: 10.5802/jtnb.187
Article PDF | Reviews MR 1469659 | Zbl 0924.11081

Résumé - Abstract

We study a generalization of the classical circle problem to real quadratic rings. Namely we study $\mathcal{C}(N,M) = \sum _{n \le N} \sum _{m \le M} r(n + m \sqrt{k})$ where $r(n + m \sqrt{k})$ is the number of representations of $n+m \sqrt{k}$ as a sum of two squares in $\mathbb{Z}[ \sqrt{k}]$ (with $k > 1$ and squarefree). Using spectral theory in $PSL_2(\mathbb{Z}) \setminus \mathbb{H}$, we get an asymptotic formula with error term for $\mathcal{C}(N, M)$, showing that some techniques on the estimation of automorphic $L$-functions can be applied to get upper bounds of the error term.

Bibliography

[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. Article |  MR 1544690 |  JFM 50.0118.01
[Lu] W. Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms, Acta Arithmetica 70 (1995), 377-391. Article |  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