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:
• Bruce C. Berndt, Junxian Li and Alexandru Zaharescu, The final problem: an identity from Ramanujan’s lost notebook, Journal of the London Mathematical Society 100 (2) (2019), 568–591.
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.
For more fun reading, try this:
• Robert P. Schneider, Uncovering Ramanujan’s “lost” notebook: an oral history.
Here’s an identity from Ramanujan’s lost notebooks: