Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

Bruce C. Berndt ; Ken Ono

Séminaire lotharingien de combinatoire, Tome 42 (1998-1999)

Résumé

When Ramanujan died in 1920, he left behind an incomplete manuscript in two parts on the partition and tau functions, which he wrote in the last year of his life. Hardy extracted a small portion of the manuscript for Ramanujan's last paper, published posthumously in 1921. Most, but not all, of the results in the manuscript have since been proved in the literature. However, the first part of the manuscript contains new congruences for the partition function. The second part of the manuscript contains a complete proof, albeit lacking in many details, of Ramanujan's conjectured congruence for the partition function modulo any power of 5. The purpose of this paper is first to present Ramanujan's complete manuscript, and second to provide details and extensive commentary for the theorems therein.