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Ramanujan observed the congruence $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$, where $\tau$ is the Ramanujan $\tau$-function. Does anybody know how he proved it, or would anybody venture an educated guess?

I know there is a proof in, accessible to Ramanujan's techniques, and a more conceptual proof by Swinnerton-Dyer that was certainly not accessible to Ramanujan. But I haven't found any proofs claiming to be the original (if indeed he had one).

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I think the Ramanujan's proof is given in the article you cited (Bruce C. Berndt, Ken Ono, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary). J. R. Wilton in his article "On Ramanujan's Arithmetical Function $\Sigma_{r,s}(n)$" writes:

"This relation follows immediately from equation (l-63) of Ramanujan's posthumous paper, Congruence properties of partitions, Math. Zeitschrift, 9(1921), 147-153 (Collected Papers, No. 30), but Ramanujan does not seem to have noticed the fact. The same formula (1-63) occurs as line 6 of Table I of the paper On certain arithmetical functions, No. 18."

As Berndt&Ono publication shows, in fact Ramanujan was well aware of this fact. I do not have access to Ramanujan's mentioned papers but I think the equation (l-63) mentioned by Wilton should be the formula (9.1) from the Berndt&Ono's article.

G.H. Hardy in his book "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" also states that it was Ramanujan who proved the congruence and refers to the article of G.N. Watson "Uber Ramanujansche Kongruenzeigenschaften der Zerfällungsan zahlen" for elaboration of Ramanujan's results.

An interesting history of discovery of the Ramanujan's "lost" notebook is described in (R.P. Schneider, Uncovering Ramanujan's "Lost" Notebook: An Oral History).

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