Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I remember having read a criticism thereof later, but remain ignorant of what happened since then. A quick googling only brings out blog articles that don't seem to give real news on its status. So, has the heuristics presented in the considered paper been comforted by some further work or not ?
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asked Dec 4, 2017 at 21:49
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2$\begingroup$ I haven't been tracking that particular paper, in "Quanta", as I recall, but I remember that someone on this site (I'm very sorry I've forgotten who), pretty-decisively explained why it cannot possibly be correct, for at least two different reasons. So then we're done. (Anyone remember who was provoked to submit something to arXiv refuting the argument?) $\endgroup$– paul garrettCommented Dec 4, 2017 at 21:59
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3$\begingroup$ See this question mathoverflow.net/questions/266935/… $\endgroup$– StoppleCommented Dec 4, 2017 at 22:14
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2$\begingroup$ Here is the paper in question: arxiv.org/abs/1704.02644 It seems the thread in question (with the refuting explanations) has been moved (?) to MSE: math.stackexchange.com/questions/2211278/… $\endgroup$– M.G.Commented Dec 4, 2017 at 22:14
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2$\begingroup$ Ah, yes, Jean Bellisard deserves credit for taking the trouble to decisively explain the problems. (Thanks for finding that! @Stopple...) $\endgroup$– paul garrettCommented Dec 4, 2017 at 22:15
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$\begingroup$ ... and @July (two comments to be able to ping both of you). $\endgroup$– paul garrettCommented Dec 4, 2017 at 22:16
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2 Answers
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It seems the only development so far is a response from Bender, Brody and Müller from 18 May 2017 to Bellissard's criticism of their work, also to be found on arxiv:
Comment on 'Comment on "Hamiltonian for the zeros of the Riemann zeta function" '
They seem to be addressing all the issues pointed out by Bellissard, although I don't know if it's to a satisfactory level.
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A new paper in Physical Review Letters on related approaches:https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.241602
Also, a summary appears in Physics :https://physics.aps.org/articles/v14/s157
answered Dec 10, 2021 at 19:40