Congruences for the Ramanujan function and generalized class numbers

Heim, Bernhard

doi:10.1090/S0025-5718-08-02136-4

Congruences for the Ramanujan function and generalized class numbers
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Math. Comp. 78 (2009), 431-439 Request permission

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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