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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Ramanujan congruences for modular forms of integral and half-integral weights
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by B. Datskovsky and P. Guerzhoy
Proc. Amer. Math. Soc. 124 (1996), 2283-2291
DOI: https://doi.org/10.1090/S0002-9939-96-03334-5

Abstract:

In 1916 Ramanujan observed a remarkable congruence: $\tau (n)\equiv \sigma _{11}(n) \quad \bmod 691$. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight $12$ and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number $B_{12}$. In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.
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Bibliographic Information
  • B. Datskovsky
  • Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
  • Email: bdats@euclid.math.temple.edu
  • P. Guerzhoy
  • Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
  • Address at time of publication: Fakultät für Mathematik und Informatik, Universität Mannheim, D-6800 Mannheim 1, Germany
  • Email: pasha@techunix.technion.ac.il, pasha@euklid.math.uni-mannheim.de
  • Received by editor(s): May 15, 1994
  • Additional Notes: The first author’s research was supported by a Fulbright fellowship.
  • Communicated by: Dennis A. Hejhal
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 2283-2291
  • MSC (1991): Primary 11F33; Secondary 11F30, 11F32, 11F37
  • DOI: https://doi.org/10.1090/S0002-9939-96-03334-5
  • MathSciNet review: 1327004