Timeline for On the Absolute Value of the Riemann Zeta Function on the Critical Line
Current License: CC BY-SA 4.0
8 events
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| May 5, 2020 at 3:56 | comment | added | GH from MO | @QuoteDave: That conjectured bound is a moment bound (which implies pointwise bounds though, see Titchmarsh's book). Pointwise bounds of the form $|\zeta(1/2+it)|<c(1+|t|)^{1/6}$ were proven with explicit constants $c>0$. I don't have time to look up this part of the literature, just use Google, MathSciNet etc. I gave you pointers. At any rate, it is a more substantial question how the $1/6$ can be reduced, so analytic number theorists are mainly occupied with that. The current record is by Fields medalist Bourgain from 2017 as recorded in the Wikipedia page. Discussion is closed on my part. | |
| May 5, 2020 at 3:49 | comment | added | DUO Labs | No, I am really only interested in $k=1$, which has been proven, and the Wikipedia page never gave any values of $c$. | |
| May 5, 2020 at 3:46 | comment | added | GH from MO | @QuoteDave: That asymptotic formula is only conjectured. As the Wikipedia page says, a conjectured formula for $c_{k,j}$ was given by Keating & Snaith (2000). The best unconditional upper bounds appear in the table in the Wikipedia page. Please study the literature. It is vast and easy to find. | |
| May 5, 2020 at 3:41 | comment | added | DUO Labs | Though when they say "$c_{k,j}$ as some constant", what kind of constants are they talking about? | |
| May 5, 2020 at 3:38 | comment | added | GH from MO | @QuoteDave: There are hundreds of papers on the subject. As a starter, read Titchmarsh's chapter and the Wikipedia entry (in my post). See also the recent work of Soundararajan, Harper etc. on the arXiv. This is one the hot topics of analytic number theory. | |
| May 5, 2020 at 3:36 | history | edited | GH from MO | CC BY-SA 4.0 |
added 133 characters in body
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| May 5, 2020 at 3:35 | comment | added | DUO Labs | Are there any unconditional results? | |
| May 5, 2020 at 3:34 | history | answered | GH from MO | CC BY-SA 4.0 |