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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.

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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 Nov 21 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 Nov 21 at 19:51
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    $\begingroup$ one formula of Ramanujan was only proven in 2001 $\endgroup$ – Carlo Beenakker Nov 21 at 20:37
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    $\begingroup$ does the Brocard-Ramanujan problem qualify? $\endgroup$ – Carlo Beenakker Nov 21 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 Nov 21 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.

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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 Nov 28 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 Nov 29 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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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 Nov 22 at 21:45
  • $\begingroup$ @JohnBaez: Enlightening and reassuring, thank you! $\endgroup$ – GH from MO Nov 22 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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