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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 Ramanujan. However, right now I'm mainly interested in formulas, i.e. identities, that he wrote down.

GH from MO
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John Baez
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    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 – Wojowu Nov 21 '20 at 19:43
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    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? – Wojowu Nov 21 '20 at 19:51
  • In some comment on MO I saw the $2^x, 3^x$ problem attributed to Ramanujan, but I can't find it now. – John Baez Nov 21 '20 at 20:03
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    one formula of Ramanujan was only proven in 2001 – Carlo Beenakker Nov 21 '20 at 20:37
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    does the Brocard-Ramanujan problem qualify? – Carlo Beenakker Nov 21 '20 at 21:11
  • @Wojowu In the paper "Highly composite numbers" point 36 in page 114 of Ramanujan's Collected Papers, it is said that the quotient of two consecutive superior highly composite numbers is a prime. This is not proved, but follows it the problem have a positive answer. I think this is not known. – juan Nov 21 '20 at 22:23
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    @juan "...follows if the problem has a positive answer." Follows if which problem has a positive answer? – Gerry Myerson Nov 21 '20 at 23:01
  • @CarloBeenakker - the Brocard-Ramanujan problem is a nice unproved conjecture of Ramanujan - thanks! But it's not a "formula". – John Baez Nov 22 '20 at 17:27
  • @GerryMyerson I was confused. It is true that there is an unproved assertion of Ramanujan at the point I referred but it is unrelated to the $2^x$, $3^x$ question. – juan Nov 23 '20 at 17:34
  • Related: Percentage of Ramanujan's conjectures that were proven correct. – Timothy Chow Nov 23 '20 at 18:09
  • The rather sad part here is that most of the proofs which are offered for the difficult identities of Ramanujan are mostly like verification based on modular forms (and use software like maple, macsyma etc). I believe there are many formulas which are not proven via pen and paper (Ramanujan used chalk and slate for the same mostly). – Paramanand Singh Dec 03 '20 at 10:55
  • And there is a lot of mystery about his methods. It is a tragedy that his mentor Hardy was no so much interested in the topic of theta functions, modular forms etc and perhaps therefore did not bother to discuss his methods related to his theory of theta functions. – Paramanand Singh Dec 03 '20 at 10:58

5 Answers5

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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.

IJL
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John Baez
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  • 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? – Timothy Chow Nov 28 '20 at 14:18
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    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. – John Baez Nov 29 '20 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.

Timothy Chow
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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 infinity 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.

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

lamlame
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    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: https://johncarlosbaez.wordpress.com/2020/11/18/ramanujans-easiest-formula/ – John Baez Nov 22 '20 at 21:45
  • @JohnBaez: Enlightening and reassuring, thank you! – GH from MO Nov 22 '20 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.

David Roberts
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Turbo
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