Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Congruences for the Ramanujan function and generalized class numbers

Author: Bernhard Heim
Journal: Math. Comp. 78 (2009), 431-439
MSC (2000): Primary 11F33, 11F67, 11F80; Secondary 11Y70
Published electronically: May 20, 2008
MathSciNet review: 2448715
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Ramanujan $ \tau$-function satisfies well-known congruences modulo the so-called exceptional prime numbers $ 2,3,5,7,23,691$. In this paper we prove new congruences related to the irregular primes $ 131$ and $ 593$, involving generalized class numbers. As an application we obtain distribution results. We obtain a new proof of the famous $ 691$ congruence and congruences of the related Rankin L-funtion.

References [Enhancements On Off] (What's this?)

  • 1. Henri Cohen, Sums involving the values at negative integers of 𝐿-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. MR 0382192,
  • 2. P. Deligne: Formes modulaires et representations l-adiques. Seminaire Bourbaki No. 355 (1969).
  • 3. Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735
  • 4. Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. MR 881269,
  • 5. Paul B. Garrett, On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients, Invent. Math. 107 (1992), no. 3, 453–481. MR 1150599,
  • 6. G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; Macmillan Company, New York, 1940. MR 0004860
  • 7. B. Heim: On the Spezialschar of Maass. Preprint, submitted 2006
  • 8. Hans Maass, Die Fourierkoeffizienten der Eisensteinreihen zweiten Grades, Mat.-Fys. Medd. Danske Vid. Selsk. 34 (1964), no. 7, 25 pp. (1964) (German). MR 0171758
  • 9. Ken Ono, Congruences on the Fourier coefficients of modular forms on Γ₀(𝑁), The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 93–105. MR 1284053,
  • 10. Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823
  • 11. S. Ramanujan: On certain arithmetical functions. Trans. Cambridge Phil. Soc. 22 (1916), 159-184.
  • 12. J.-P. Serre: Congruences et formes modulaires (d'apres Swinnerton-Dyer). Seminaire Bourbaki No. 416 (1971).
  • 13. D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 105–169. Lecture Notes in Math., Vol. 627. MR 0485703

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11F33, 11F67, 11F80, 11Y70

Retrieve articles in all journals with MSC (2000): 11F33, 11F67, 11F80, 11Y70

Additional Information

Bernhard Heim
Affiliation: Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Received by editor(s): November 13, 2007
Received by editor(s) in revised form: January 9, 2008
Published electronically: May 20, 2008
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society