Timeline for Quasicrystals and the Riemann Hypothesis
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 25, 2013 at 3:02 | review | First posts | |||
| Jun 25, 2013 at 6:17 | |||||
| Jun 13, 2013 at 4:10 | comment | added | Broadeducation | There are two papers of Bombieri where the relevance of the explicit formula to the truth of the Riemann Hypothesis is investigated. Here they are: "A variational approach to the explicit formula", Comm. Pure Appl. Math. 56 (2003), no. 8, 1151–1164. and "Remarks on Weil's quadratic functional in the theory of prime numbers. I.", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 3, 183–233 (2001). | |
| Jun 13, 2013 at 4:04 | comment | added | Broadeducation | ... I need to think more about it but certainly most versions of the explicit formula that I came accross assumes that at least one of $h$ or $\widehat{h}$ is analytic in some strip, since complex analysis is used in the proof. The only exception is arxiv.org/abs/1203.5328 (section 2) where the explicit formula is proven by a rather convoluted method, but it might look a bit more like what you would like to do... | |
| Jun 13, 2013 at 4:02 | comment | added | Broadeducation | The way I see it your equation above is a suggestive and heuristic way of writting the explicit formula. Certainly if you integrate against certain smooth functions you will recover (more or less) the explicit formula (some terms seems to be missing in the formula above). However I don't think that the explicit formula is true for any smooth test function, and in particular this might suggest that the distributional formula above is not exactly "true 100% of the time"... | |
| Jun 13, 2013 at 2:33 | comment | added | John Baez | By the way, I don't see how to derive the equation I wrote down from the one in Lemma 1. I haven't tried very hard, but I'd like to be assured it's possible. I'd need to see what happens with the term involving the logarithmic derivative of the gamma function. | |
| Jun 13, 2013 at 1:48 | comment | added | John Baez | I think that helps a lot. If the Riemann Hypothesis holds, all the $k_j$ are real, so the Fourier transform of the right-hand side will be a linear combination of the Dirac deltas. If the Riemann Hypothesis is false, some of the $k_j$ will be complex, so I see no reason to expect the Fourier transform of the right-hand side to be a linear combination of Dirac deltas. However, it would take me some work to prove it's not. Has someone done that? | |
| Jun 13, 2013 at 1:21 | history | answered | Broadeducation | CC BY-SA 3.0 |