Timeline for Approaches to Riemann hypothesis using methods outside number theory [closed]
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39 events
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| Jun 30, 2019 at 0:54 | comment | added | mike | In my opinion, "methods outside of number theory" means that they do not use any arithmetic functions like Mobius function $\mu(n)$. I would like to mention the Polya-Hurwitz approach. Please see the 2017 preprint by Shi (arxiv.org/abs/1706.08868). By truncating the Fourier Kernel $\Phi(t)$ and Fourier transformation integration range, Shi constructed a family of functions $\{F(n,z)\}_{n=9}^{\infty}$ that uniformly converge to the Riemann $\Xi(z)$ function in the critical strip $|Im(z)|<1/2$; Shi then proved that all the zeros of $W(n,z)$, a variant of $F(n,z)$, are real. | |
| Sep 11, 2013 at 13:04 | review | Reopen votes | |||
| Sep 11, 2013 at 13:06 | |||||
| Dec 14, 2012 at 21:39 | history | closed |
Felipe Voloch user9072 Andy Putman Asaf Karagila♦ Yemon Choi |
no longer relevant | |
| Dec 14, 2012 at 21:39 | comment | added | Yemon Choi | As per tea.mathoverflow.net/discussion/1488/… , casting a vote to close as "no longer relevant". | |
| Oct 12, 2010 at 16:53 | comment | added | Unknown | Here is Iwaniec defending Analytic Number Theory at the end of his talk:mathunion.org/ICM/ICM2006.1/Main/… | |
| Aug 22, 2010 at 4:29 | answer | added | anon | timeline score: 10 | |
| Aug 7, 2010 at 6:23 | comment | added | Emerton | Just to give one defense of Hardy (who surely regarded himself as a number theorist) against Weil's charge (not that one is needed; his legacy can speak for itself), let me remark that his proof of the existence of an infinitude of zeroes of the zeta function lying on the critical line uses the Mellin transform relationship between the zeta function and the Jacobi theta function (a relationship introduced by Riemann, by the way!), and then studies the question on the automorphic side (so to speak). I don't think that you can get into much deeper number theoretic territory than this. | |
| Aug 7, 2010 at 5:51 | comment | added | Chandan Singh Dalawat | "... there is a subject in mathematics (it's a perfectly good and valid subject and it's perfectly good mathematics) which is misleadingly called Analytic Number Theory. In a sense it was born with Riemann who was definitely not a number-thorist; it was carried on, among others, by Hadamard, and later by Hardy, who were also not number-thorists (I knew Hadamard well); and to the best of my understanding analytic number theory is not number theory." Weil, Two Lectures, 1974. | |
| Aug 7, 2010 at 2:47 | comment | added | Felipe Voloch | You have a very narrow view of what constitutes number theory. | |
| Aug 7, 2010 at 1:35 | comment | added | Theo Johnson-Freyd | @Victor Protsak: for future reference, the link is mathoverflow.net/faq#openproblems . We should probably ask Anton to add that to the table of contents at the FAQ. | |
| Aug 7, 2010 at 0:41 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 6, 2010 at 22:31 | history | edited | Anweshi | CC BY-SA 2.5 |
For the true reason names must remain.
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| Aug 6, 2010 at 22:29 | history | rollback | Anweshi |
Rollback to Revision 7
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| Aug 6, 2010 at 22:25 | history | edited | Victor Protsak | CC BY-SA 2.5 |
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| Aug 6, 2010 at 21:46 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 6, 2010 at 21:29 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 6, 2010 at 21:06 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 6, 2010 at 18:01 | history | edited | Anweshi |
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| Aug 6, 2010 at 16:34 | comment | added | Anweshi | Nothing in the book of Ivic or Titchmarsch and Heath-Brown. More precisely, methods outside the traditional sybjects of elementary number theory and analytic number theory. I have given two examples above. One with algebraic geometry and one with thermodynamics. | |
| Aug 6, 2010 at 16:09 | answer | added | doetoe | timeline score: 6 | |
| Aug 6, 2010 at 13:56 | comment | added | stankewicz | To ask a revised version of Tom Smith's question: What exactly would qualify as a non-number-theoretic method? | |
| Aug 6, 2010 at 13:51 | history | made wiki | Post Made Community Wiki by Anweshi | ||
| Aug 6, 2010 at 13:35 | comment | added | Anweshi | I just asked for an approach that a person answering might think to be promising. We do not want to hear just every failes attempt, do we? I do not want to give the open problem tag because I am uncomfortable with the connotation that I am asking to prove the Riemann hypothesis. | |
| Aug 6, 2010 at 8:30 | answer | added | Peter Arndt | timeline score: 14 | |
| Aug 6, 2010 at 8:01 | answer | added | Charles Matthews | timeline score: 12 | |
| Aug 6, 2010 at 4:24 | comment | added | Victor Protsak | Shouldn't this be tagged "open-problem" and made CW, per FAQ? Also, in the absence of a proof, whether a method is viewed as promising or not is highly subjective. | |
| Aug 6, 2010 at 3:11 | answer | added | Pete L. Clark | timeline score: 23 | |
| Aug 6, 2010 at 2:18 | answer | added | Felipe Voloch | timeline score: 25 | |
| Aug 6, 2010 at 1:08 | answer | added | Micah Milinovich | timeline score: 5 | |
| Aug 6, 2010 at 0:29 | comment | added | Anweshi | @Tom Smith: Yes. Though I didn't mention it explicitly, what the professor told me was that they failed completely hopelessly. Whereas the non-number theoretic approaches require some theory-building and there is hope yet and for the meantime we try to do the groundwork. | |
| Aug 6, 2010 at 0:12 | comment | added | Tom Smith | "All attempts to prove the Riemann hypothesis using number theoretic methods have failed". Isn't this just a special case of "all attempts to prove the Riemann hypothesis have failed"? | |
| Aug 5, 2010 at 23:06 | comment | added | Anweshi | @David Hansen: I mean that even if they aren't successful they might unearth a lot of interesting math. Indeed, the proof of Weil conjectures is already a lot of interesting math. | |
| Aug 5, 2010 at 23:05 | comment | added | David Hansen | I personally don't believe that any proposed approach to the Riemann hypothesis, including the two you listed, deserve to be called "promising". | |
| Aug 5, 2010 at 23:00 | answer | added | Joseph O'Rourke | timeline score: 10 | |
| Aug 5, 2010 at 22:57 | comment | added | Anweshi | If you mean Random matrices, no, it is a separate approach. | |
| Aug 5, 2010 at 22:56 | comment | added | Alon Amit | Is the "hidden operator" approach initiated by Dyson and Montgomery subsumed in the the Bost-Connes approach? en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture | |
| Aug 5, 2010 at 22:54 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Aug 5, 2010 at 22:53 | history | edited | Evan Jenkins |
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| Aug 5, 2010 at 22:50 | history | asked | Anweshi | CC BY-SA 2.5 |