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Are there any formulas due to Ramanujan that have still not been proved—or disproved?

If so, what are they?

I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^x$ and $3^x$ are both integers then $x$ is an integer. There may be other open conjectures due to Ramanujan. However, right now I'm mainly interested in formulas, i.e. identities, that he wrote down.

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    $\begingroup$ The Ramanujan conjecture for the tau function (and other holomorphic cusp forms) has been proven by Deligne (and Serre in the weight 1 case). There are extensions of these conjectures which are unproven (like for Maass forms) but those were not posed by Ramanujan $\endgroup$
    – Wojowu
    Commented Nov 21, 2020 at 19:43
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    $\begingroup$ The $2^x,3^x$ problem is one of the most popular questions on all of MO, which I would take as (perhaps weak) indication of it being open. Do you happen to have a reference to Ramanujan posing this question? $\endgroup$
    – Wojowu
    Commented Nov 21, 2020 at 19:51
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    $\begingroup$ one formula of Ramanujan was only proven in 2001 $\endgroup$
    – Carlo Beenakker
    Commented Nov 21, 2020 at 20:37
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    $\begingroup$ does the Brocard-Ramanujan problem qualify? $\endgroup$
    – Carlo Beenakker
    Commented Nov 21, 2020 at 21:11
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    $\begingroup$ @juan "...follows if the problem has a positive answer." Follows if which problem has a positive answer? $\endgroup$
    – Gerry Myerson
    Commented Nov 21, 2020 at 23:01

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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.

EDIT: However, only on December 21st, 2021 did Örs Rebák submit this paper to the arXiv:

in which he completed an incomplete formula in Ramanujan's lost notebook, and proved it. So there may still be gems left to polish.

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  • $\begingroup$ So Berndt doesn't consider the Brocard-Ramanujan problem to be a "remaining conjecture" of Ramanujan, I guess? Or maybe he was considering only "formulas" because you were limiting yourself to formulas? $\endgroup$
    – Timothy Chow
    Commented Nov 28, 2020 at 14:18
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    $\begingroup$ I told Berndt "There is at least one unproved conjecture with Ramanujan's name on it, though." He said "I am unaware of (or most likely forgot) an outstanding Ramanujan conjecture." I mentioned the Brocard-Ramanujan problem and said "I see you've worked on it! But maybe this doesn't count as a conjecture due to Ramanujan: Wikipedia says Erdos posed it as a conjecture." He told me where to read more about it. But he didn't say whether he considers it a remaining conjecture of Ramanujan. $\endgroup$
    – John Baez
    Commented Nov 29, 2020 at 20:55
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Bruce Berndt has claimed that all the claims in Ramanujan's "Lost Notebook" have been proved, with a solution to the the final problem being published by Berndt, Li, and Zaharescu in J. London Math. Soc. in 2019. However, I am not sure that this means that all the formulas in Ramanujan's other writings have been proved. If you have not yet tried directly writing to Bruce Berndt, that would be my first suggestion.

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As far as I know, at least the following Ramanujan's claim about mock theta functions has not been proved, which appeared in a letter from Ramanujan to Hardy. Ramanujan claimed that: Let $q=e^{-t}$, then one has an asymptotic expansion of form \begin{align}1+\frac{q}{(1-q)^2}+&\frac{q^{3}}{(1-q)^2(1-q^2)^2}+\frac{q^{6}}{(1-q)^2(1-q^2)^2(1-q^3)^2}+\cdots\\ &=\sqrt{\frac{t}{2\pi\sqrt{5}}}\exp\left(\frac{\pi^2}{5t}+\frac{t}{8\sqrt{5}}+a_2t^2+a_3t^3+\cdots+O(a_kt^k)\right),\; t\rightarrow 0^+, \end{align} with infinitely many $a_k\neq 0$. See pages 57-58 of [Watson, G. N. The Final Problem : An Account of the Mock Theta Functions. J. London Math. Soc. 11 (1936), no. 1, 55–80.] for details.

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I don't have enough reputation to comment. Has this expression for $\sqrt{\pi e^x/2x}$ been proved?

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    $\begingroup$ Yes, that formula has been proved. I gave a talk about that formula the day before yesterday, and a listener raised the question that I'm asking here. Hardy called this formula one of Ramanujan's "least impressive" identities, and the proof goes back to Jacobi. My talk explains the proof, which is actually very pleasant: johncarlosbaez.wordpress.com/2020/11/18/… $\endgroup$
    – John Baez
    Commented Nov 22, 2020 at 21:45
  • $\begingroup$ @JohnBaez: Enlightening and reassuring, thank you! $\endgroup$
    – GH from MO
    Commented Nov 22, 2020 at 22:46
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Manjul Bhargava says only half done around 54:39 of the video Manjul Bhargava, Steven Strogatz, Matt Brown and Lynn Sherr — The Infinite Mind from March 2016.

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