Gaurav BhatnagarAffiliation: RamanujanExplained.org Title Of Talk: Ramanujan and log derivatives Abstract: The recurrence $$ np(n) = \sum_{i=1}^{n}\sigma(i)p(n-i) $$ has been used by Erd\H{o}s and credited to Ford (1931) but appears in Ramanujan's notebooks. Here the divisor function $\sigma(n)$ is related to the partition function $p(n)$. We show how Ramanujan could have obtained this result, and show some evidence that this was one of Ramanujan's standard tricks. Using a slight modification of this trick, we obtain an elementary proof of Ramanujan's famous congruences $p(5n+4) \equiv 0 \pmod 5$ and $\tau(5n+5) \equiv 0 \pmod 5$. The proof requires no more than what Euler and Jacobi (and Ramanujan) knew and embeds these results in an infinite set of such congruences. This is joint work with Hartosh Singh Bal.
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Last update made Tue Mar 10 21:28:01 CDT 2026.
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