Ramanujan’s Congruences Modulo Powers of 5, 7, and 11 Revisited

When/Where:

March 18, 2014, at 1:55pm, in LIT 368

Abstract:

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 non-negative 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.