Today I read the following brief but insightful account of Ramanujan's approach to mathematics: https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf and while reading this I wondered whether we have a lower-bound on the percentage of Ramanujan's conjectures which are correct.

I'm planning to get a copy of Ramanujan's notebooks. Meanwhile, the above question intrigues me.


This interview with Prof. Bruce Berndt indicates the percentage of correct results from his notebooks to be greater than 99.7%. (See also this longer writeup.)

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. I have gone through every entry in the notebooks. If a result has already been proved in the literature, then I just wrote the entry down and said that proofs can be found in this literature and so on.There are a number of misprints. I did not count the number of serious mistakes but it is an extremely small number - maybe five or ten out of over 3,000 results. Considering that Ramanujan did not have any rigorous training, it is really amazing that he made so few mistakes.

Bruce Berndt, Ramanujan's Notebooks, parts I--V.

side question: The Ramanujan–Petersson conjecture for Maass forms is still open.

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    $\begingroup$ Of course the Ramanujan-Petersson conjecture for Maas forms is not actually Ramanujan's per se. $\endgroup$ – Josiah Park Dec 18 '18 at 13:52
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    $\begingroup$ Do you really mean 0.997%? I guess that should read 99.7% of correct theorems. $\endgroup$ – Manfred Weis Dec 18 '18 at 14:03
  • $\begingroup$ thank you @ManfredWeis , for correcting my silly typo. $\endgroup$ – Carlo Beenakker Dec 18 '18 at 14:48
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    $\begingroup$ "really amazing" - is that one of those understatements mathematicians are fond of? $\endgroup$ – davidbak Dec 18 '18 at 16:02

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