Timeline for Bound on $L^2$ norm of $1/\zeta(1+i t)$?
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15 events
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| Sep 11, 2021 at 4:55 | comment | added | H A Helfgott | Ah, it isn't (per the other answer). | |
| Sep 10, 2021 at 18:34 | comment | added | H A Helfgott | Well, I am not, so I'm still trying to see whether there is an alternative method :). Do you think the use of zero-density results is essential? | |
| Sep 10, 2021 at 17:34 | comment | added | Terry Tao | Sounds like you can take it from here then. (If you were one of my graduate students, this would be the point where I would suggest that you work on it and report back in a week.) | |
| Sep 10, 2021 at 17:19 | comment | added | H A Helfgott | Hah. I really mean that there are methods for which one can see that explicit work should be straightforward, and methods for which explicit work is most likely going to be a nightmare that self-respecting people do not put themselves through. I am still hoping for something in the first category. (People don't think of explicit work as "clean", but in fact it forces one to streamline.) | |
| Sep 10, 2021 at 17:15 | comment | added | Terry Tao | I'm sure you are more than qualified to do so, Harald. | |
| Sep 10, 2021 at 17:07 | comment | added | H A Helfgott | ... and one that one might hope to make explicit and rigorous, with reasonable constants. | |
| Sep 10, 2021 at 17:03 | comment | added | Terry Tao | Heuristically, $\zeta(1+it) = \prod_p (1-p^{-1-it})^{-1}$ should behave statistically like $\prod_p (1-p^{-1} e^{2\pi i\theta_p})^{-1}$ for iid random variables $\theta_p \in {\bf R}/{\bf Z}$. So the asymptotic constant $c$ should be $\prod_p \int_0^1 |1-p^{-1} e^{2\pi i\theta}|^{2}\ d\theta$. By Plancherel this simplifies to $\prod_p (1+\frac{1}{p^2}) = \zeta(2)/\zeta(4) = 15/\pi^2$. Perhaps there is a more elementary way to reach this asymptotic. | |
| Sep 10, 2021 at 16:23 | comment | added | H A Helfgott | I agree. What I'd still like to see is a clean way to obtain an asymptotic. | |
| Sep 10, 2021 at 16:22 | comment | added | Terry Tao | That would have been worth noting in the OP then :-). I've just added a remark here that the method described here in fact yields asymptotics for all fractional moments, not just the -2 moment; it seems difficult to replicate that level of generality with a Plancherel-based argument. | |
| Sep 10, 2021 at 16:20 | history | edited | Terry Tao | CC BY-SA 4.0 |
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| Sep 10, 2021 at 15:49 | history | edited | Terry Tao | CC BY-SA 4.0 |
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| Sep 10, 2021 at 15:45 | comment | added | H A Helfgott | I'm almost positive that one can get an upper bound of $O(T)$ in a much simpler way from Plancherel. Let me see. Let us also look up Laurincikas - and of course I'd be thrilled to see an asymptotic already in the literature. | |
| Sep 10, 2021 at 15:18 | history | edited | Terry Tao | CC BY-SA 4.0 |
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| Sep 10, 2021 at 15:13 | history | edited | Terry Tao | CC BY-SA 4.0 |
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| Sep 10, 2021 at 15:07 | history | answered | Terry Tao | CC BY-SA 4.0 |