Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Bruce C. Berndt, Ken Ono
Ramanujan's Unpublished Manuscript on the Partition and Tau Functions
with Proofs and Commentary
Abstract.
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.
Received: December 17, 1998; Accepted: February 25, 1999.
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