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"During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known." (Source)

What are some of Ramanujan's false claims?

Note: A long time ago, I remember reading a Hardy quote saying something to the effect that where Ramanujan was wrong, he was likely right in some "deeper" sense. I assume this was referring to claims such as

$$1+2+3+4+⋯=-\frac{1}{12}$$

I do not consider this statement a "false claim", but rather an imprecise way of saying that the analytic continuation of the zeta function evaluates to $-1/12$ at $-1$.

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When Ramanujan contacted Hardy he thought that he could provide a (nearly) exact formula for $\pi(n)$ the prime counting function. This formula implicitely supposed that $\zeta$ had no complex zero at all! This is neatly detailed by Hardy at the beginning of his book "Ramanujan". See too Berndt's "Ramanujan and the theory of prime numbers" for some details.

Raymond Manzoni
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