Theorems on partitions from a page in Ramanujan's lost notebook

摘要

On page 189 in his lost notebook, Ramanujan recorded five assertions about partitions. Two are famous identities of Ramanujan immediately yielding the congruences p(5n+4)≡0(mod5) and p(7n+5)≡0(mod7) for the partition function p(n). Two of the identities, also originally due to Ramanujan, were rediscovered by M. Newman, who used the theory of modular forms to prove them. The fifth claim is false, but Ramanujan corrected it in his unpublished manuscript on the partition and τ-functions. The purpose of this paper is to give completely elementary proofs of all four claims. In particular, although Ramanujan's elementary proof for his identity implying the congruence p(7n+5)≡0(mod7) is sketched in his unpublished manuscript on the partition and τ-functions, it has never been given in detail. This proof depends on some elementary identities mostly found in his notebooks; new proofs of these identities are given here.