Timeline for Hardy-Littlewood vs heuristics on the zeta zeros
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Oct 6, 2018 at 7:22 | history | edited | user129745 | CC BY-SA 4.0 |
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| Oct 5, 2018 at 19:16 | history | edited | Greg Martin | CC BY-SA 4.0 |
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| Oct 5, 2018 at 16:03 | comment | added | Terry Tao | Actually, Rodgers' thesis has results closer to what you are looking for (uniform versions of H-L imply partial verification of GUE). escholarship.org/uc/item/2zz111pj | |
| Oct 5, 2018 at 16:01 | comment | added | Terry Tao | On the other hand, the pair correlation conjecture is known by work of Goldston and Montgomery to be equivalent to a variance asymptotic for the PNT in short intervals, which in turn can be partially derived from the H-L conjecture. See for instance this article of Rodgers mast.queensu.ca/~br66/ArithmeticGUE.pdf for more discussion. | |
| Oct 5, 2018 at 16:01 | comment | added | Terry Tao | Heuristically, the Hardy-Littlewood conjecture is linked to the GUE hypothesis on the zeta zeroes. There is a non-rigorous demonstration of this connection by Bogolmony and Keating, but getting a precise rigorous connection is difficult (one needs very strong bounds on error terms in the H-L conjecture, plus additional cancellation in these terms, to get full GUE). | |
| Oct 5, 2018 at 13:47 | comment | added | reuns | Don't forget the $k$ in $\int_2^x\frac{dt}{\log^{k} (t)}$ (if there are no obstructions). $X_n = 1_{n \text{ is prime}}$ would then look like a sequence of (locally) independent random variables. If in $\frac{-\zeta'}{\zeta}(s) =\sum_n \Lambda(n) n^{-s}$ you switch $\Lambda(4n+1)$ and $\Lambda(4n+2)$ it won't change the location of the zeros in the critical strip (there will be new ones in $\Re(s) <0$) thus you can hardly see the twin prime conjecture in the zeros. | |
| Oct 5, 2018 at 10:26 | history | edited | user129745 | CC BY-SA 4.0 |
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| Oct 5, 2018 at 10:21 | history | edited | user129745 | CC BY-SA 4.0 |
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| Oct 5, 2018 at 10:15 | history | edited | user129745 | CC BY-SA 4.0 |
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| Oct 5, 2018 at 10:15 | review | First posts | |||
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| Oct 5, 2018 at 10:10 | history | asked | user129745 | CC BY-SA 4.0 |