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While trying to formulate an answer to this question, I realized I really have no idea how Ramanujan came up with his formulas. Bruce Berndt has a number of great expository articles, e.g., this one, but I couldn't discern how Ramanujan approached problems. There are famous stories about how solutions seemed to just pop into his head, e.g., when he quickly solved a tricky problem and was asked how, he responded:

It was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind.

According to Wikipedia, Hardy said Ramanujan's results were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." In the same article, Ramanujan is quoted as saying "An equation for me has no meaning unless it expresses a thought of God" and crediting his mathematical abilities to his family goddess Namagiri Thayar.

Previous MathOverflow questions have asked how he came up with specific results, and in this mathoverflow question, Tim Chow said "Ramanujan is legendary for having an extraordinary, uncanny intuition, and it is natural to try to understand this intuition better."

Question: Now that so many of Ramanujan's formulas have been verified, that his notebooks have been carefully studied, and that his results have been understood as a part of a larger theory, has anyone discerned a pattern or a set of standard tricks/approaches that might have been underlying how he came up with his results?

What I've read from Berndt suggests that Ramanujan's work focused heavily on continued fractions, partition functions, asymptotic formulas, modular forms, zeta functions, $q$-series, Eisenstein series, and mock theta functions. I'd be happy for an answer in any of these individual areas. Berndt suggests Ramanujan worked on slate, and erased his work when finished, recording only the final formulas he discovered, so we can perhaps deduce that Ramanujan had some fairly compact way to do his work.

Side note: while it's fun to have stories of mathematicians so brilliant that no one can understand them, I don't think this is the right point of view if we want to make the field welcoming to newcomers. It's also not very satisfying from the point of view of really understanding what's going on in a field. I hope that one day the mathematical community will understand everything Ramanujan did, and now, 100 years after his death, I'm hoping there has been some progress on this goal.

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  • $\begingroup$ I debated making this CW, but the other two MO questions I linked to weren't CW, so I decided to hold off. I'm not really sure what's standard with ho.history-overview questions $\endgroup$ – David White Jun 20 at 0:31
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    $\begingroup$ Even if we understood everything he did, we will never know how he did them. Not even close.He obviously had a lot of intuition and computational/algebraic experience. Even if we found a few patterns and tricks, there is no way to know if he used them or else used alternate methods. $\endgroup$ – Somos Jun 20 at 14:42
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    $\begingroup$ @Somos I am not willing to give up so easily. I am sure that when Cardano was solving cubic equations left and right it seemed like magic, but there was a pattern and a trick to it. When Galois characterized which polynomials can be solved by radicals, it seemed like magic, but now we understand it in the context of Galois theory and teach it to undergraduates. The same with Gauss's work on the fast Fourier transform. We should aspire to truly understand Ramanujan's way of thinking, rather than assuming it's beyond our collective power. $\endgroup$ – David White Jun 20 at 14:55
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    $\begingroup$ Ramanujan had a great skill in algebraic manipulation (much better than current symbolic software). Almost all his independent (of Hardy) work is based on algebraic manipulation. And note that the processes of calculus were also a part of algebraic manipulation for him. $\endgroup$ – Paramanand Singh Jun 25 at 5:13
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    $\begingroup$ Also I think there is another factor in case of Ramanujan. He was using powers of algebraic manipulation almost all the time 24x7. Ramanujan's approach therefore also involves a great deal of labour involving highly efficient manipulation. A person who used similar approach was Jacobi. It's rather unfortunate that the techniques of Jacobi and Ramanujan have been ditched in modern times. $\endgroup$ – Paramanand Singh Jun 25 at 6:35
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This is an exposition of my comments via actual examples. I will present a few of the tricks which Ramanujan heavily used (all of these are algebraic manipulation and do not involve anything high brow).

Partial Fractions

Often Ramanujan used to derive partial fractions for many functions (usually made of circular/hyperbolic functions). He never mentioned explicitly the technique used but it appears it was based on analysis of the poles of the function. However this did not involve complex analysis and instead it was an extension of the method used for typical rational functions in such a way to avoid common pitfalls. Partial fractions were then heavily used to obtain many series by comparing coefficients. In particular his formula related to $\zeta(2n+1)$ is derived in this manner (see this thread). Another application of the technique is described here.

Multisection of Series

This involves splitting a power series into multiple series by grouping terms with powers modulo a given number $n$.

Ramanujan used the technique in a different manner by trying to analyze power series for $f(x^{1/n})$ and collecting terms which contain fractional powers of $x$. Using this approach he proved many properties of Rogers Ramanujan continued fraction and also obtained generating functions of $p(5n+4),p(7n+5)$. A nice application of this technique is presented here.

Simplification of algebraic expressions

If $x, y$ are two numbers connected by algebraic equation (in theory) of the form $P(x, y) =0$ where $P$ is a polynomial in $x, y$ with integer coefficients then Ramanujan would often try to guess simple functions like $u=f(x), v=g(y) $ so that the relation between $x, y$ could be transformed into a visually simple form as $F(u, v) =0$ where $F$ need not be polynomial but rather general algebraic function.

Here it appears that he worked by trial and error and put a lot of effort to simplify the form of the algebraic relation. This is clearly seen when one compares Ramanujan's class invariants with the corresponding ones given by Weber. His modular equations are also in much simpler form compared to those given by others.

In this connection it should also be noted that Ramanujan had discovered many algebraic identities which helped him to denest radicals. I don't think there was a technique involved here. The identities were developed in pursuit of specific goals like expressing a number as sum of two cubes in two different ways or in another case for finding simple expressions for singular moduli. Also he wasn't aware of any Galois theory and probably he did not need it. I guess he used his time and skill to figure out these by trial and error (discarding quickly anything which did not seem to meet the desired goal).


Note: Some of the examples presented above are available on Math.SE and I will add links to them after some time.

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