Ramanujan's Congruences Modulo Powers of 5, 7, and 11 Revisited
Sprache des Vortragstitels:
The number of partitions of 4 is p(4)=5, namely: 4,3+1, 2+2, 2+1+1, and 1+1+1+1. Ramanujan observed that p(5n+4) is divisible by 5 for all nonnegative integers n. More generally, Ramanujan discovered similar congruences modulo 7 and 11, and also for all powers of these primes. The cases 5 and 7 were proved by G.N. Watson (1938); in 1984, Frank Garvan was able to simply this proof significantly. In 1967 the case 11 was proved by A.O.L. Atkin; in 1983, B. Gordon presented another approach being closer to Watson's. The talk originates from joint work with Silviu Radu; it describes a new algorithmic setting in the theory of modular functions that gives rise to a new unified frame to prove Ramanujan's celebrated families of partition congruences.