When I gave a talk about Ramanujan’s easiest formula at the Whittier College math club run by my former student Brandon Coya, one of the students there asked a great question: are there any unproved formulas of Ramanujan left?

So I asked around on MathOverflow, and this is the result:

George Andrews and Bruce Berndt have written five books about Ramanujan’s lost notebook, which was actually not a notebook but a pile of notes Andrews found in 1976 in a box at the Wren Library at Trinity College, Cambridge. In 2019 Berndt wrote about the last unproved identity in the lost notebook:

Following Timothy Chow’s advice, I consulted Berndt and asked him if there were any remaining formulas of Ramanujan that have neither been proved nor disproved. He said no:

To the best of my knowledge, there are no claims or conjectures remaining. There are some statements to which we have not been able to attach meaning.

I checked to make sure that this applies to all of Ramanujan’s output, not just the lost notebook, and he said yes.

It’s sort of sad. But there’s a big difference between proving an identity and extracting all the wisdom contained in that identity! A lot of Ramanujan’s formulas have combinatorial interpretations, while others are connected to complex analysis—e.g. mock theta functions—so I’m sure there’s a lot of good work left to do, inspired by Ramanujan’s identities. There is also a continuing industry of discovering Ramanujan-like identities.

Here’s an identity from Ramanujan’s lost notebooks:

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“There are some statements to which we have not been able to attach meaning.” Here is a recent example, we link combinatorics of polytopes (Stanley order polynomials, Ehrhart theory) and zeta values to justify coefficients in a formula of Ramanujan.

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The link to arXiv in the formula’s picture has an extra parenthesis at the end.

Thanks, I’ll fix it!

“There are some statements to which we have not been able to attach meaning.”

Here is a recent example, we link combinatorics of polytopes (Stanley order polynomials, Ehrhart theory) and zeta values to justify coefficients in a formula of Ramanujan.

Cool! It’s interesting to me that a nice operad shows up here.