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The Ramanujan τ -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.

Congruences for the Ramanujan function and generalized class numbers