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Quoting his Wikipedia page (current revision):

compiled nearly 3,900 results

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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    $\begingroup$ 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. $\endgroup$ – Marek Mitros Dec 13 '17 at 10:17
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    $\begingroup$ 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. $\endgroup$ – Steven Landsburg Dec 13 '17 at 15:45
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    $\begingroup$ $\sum_{i=0}^\infty i = -{1 \over 12}$ :D $\endgroup$ – Eric Duminil Dec 13 '17 at 18:38
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    $\begingroup$ 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. $\endgroup$ – Timothy Chow Dec 13 '17 at 18:50
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    $\begingroup$ @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. $\endgroup$ – Timothy Chow Dec 14 '17 at 16:36
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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.

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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.

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