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Quoting his Wikipedia page (current revision):

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Nearly all his claims have now been proven correct

Which of his claims have been disproven, can any insight be gained from the mistakes of this genius?

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StanOverflow
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    Even genius can make mistakes. He was a human being, I agree that genius. I read that some formulas or solutions were just come from his subconscious to his mind. Even he didn't know, how it happened. It may happen to everyone that sometimes some ideas come to the mind suddenly some time after you stopped thinking on some problem. If you stopped but there is some "background job" runnning in the brain which may return results after some time. –  Dec 13 '17 at 10:17
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    If questions of the form "What did [X] get wrong?" (as opposed to, say, "What misconception led [X] to believe the specific false statement [Y]?") are on topic , I guess we should expect about 3000 such questions. – Steven Landsburg Dec 13 '17 at 15:45
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    $\sum_{i=0}^\infty i = -{1 \over 12}$ :D – Eric Duminil Dec 13 '17 at 18:38
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    I don't think that this question should be closed. This is not a random question about the mistakes of a random mathematician. Ramanujan is legendary for having an extraordinary, uncanny intuition, and it is natural to try to understand this intuition better. To do so, it makes sense to look at the times when the intuition was wrong as well as when the intuition was right. – Timothy Chow Dec 13 '17 at 18:50
  • How does that quote make you think he got something wrong? It doesn't say or imply that at all. "Wrong until proven correct", is that how you think? – Stefan Pochmann Dec 14 '17 at 11:50
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    @StefanPochmann : How does StanOverflow's question make you think that he believes that the Wikipedia quote says or implies that Ramanujan got something wrong? StanOverflow doesn't say that at all. – Timothy Chow Dec 14 '17 at 16:36
  • @TimothyChow If they didn't, then they would've asked "Did Ramanujan get something wrong?" instead of "What did Ramanujan get wrong?". That latter question implies that he did get something wrong, and they're asking what. – Stefan Pochmann Dec 14 '17 at 16:39
  • @StefanPochmann : It still doesn't follow that the Wikipedia quote was the source of the belief that Ramanujan got something wrong. – Timothy Chow Dec 14 '17 at 16:42
  • @TimothyChow If it was something else, then they would've shown that something else. – Stefan Pochmann Dec 14 '17 at 16:45
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    @StefanPochmann : Maybe, but that's a conjecture. If you're going to pick at others' nits then your own nits are vulnerable to being picked at as well. – Timothy Chow Dec 14 '17 at 19:01
  • @TimothyChow Don't see how mine are vulnerable. They make perfect sense. – Stefan Pochmann Dec 14 '17 at 19:02
  • Let us continue this discussion in chat. – Timothy Chow Dec 14 '17 at 19:15

2 Answers2

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Bruce Berndt writes,

Most of Ramanujan's mistakes arise from his claims in analytic number theory, where his unrigorous methods led him astray. In particular, Ramanujan thought his approximations and asymptotic expansions were considerably more accurate than warranted. In [12], these shortcomings are discussed in detail.

[12] is Berndt, Ramanujan's Notebooks, Part IV.

Gerry Myerson
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    This might be a good partial answer to the old MO question that asks for examples where rigor is important. https://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important – Timothy Chow Dec 13 '17 at 18:53
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Hardy wrote some things about this, as I learned when writing this blog post. Here is a mistake which was even featured in the Ramanujan movie: in his letters to Hardy, Ramanujan claimed to have found an exact formula for the prime counting function $\pi(n)$, but (in Hardy's words)

Ramanujan’s theory of primes was vitiated by his ignorance of the theory of functions of a complex variable. It was (so to say) what the theory might be if the Zeta-function had no complex zeros. His method depended upon a wholesale use of divergent series… That his proofs should have been invalid was only to be expected. But the mistakes went deeper than that, and many of the actual results were false. He had obtained the dominant terms of the classical formulae, although by invalid methods; but none of them are such close approximations as he supposed.

Based on the second sentence in particular it sounds like what happened, although I haven't checked, was that Ramanujan's formula was the explicit formula but missing the contribution from the complex zeroes of the zeta function.

Qiaochu Yuan
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