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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
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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.

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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