Timeline for Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2023 at 12:41 | history | edited | Agno | CC BY-SA 4.0 |
Further simplified equation (1)
|
| Jul 22, 2023 at 0:05 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 111 characters in body
|
| Jul 20, 2023 at 17:22 | comment | added | Stopple | There is work by Farr-Pauli-Saidak on zeros of fractional derivatives of $\zeta(s)$, see for example link.springer.com/chapter/10.1007/978-3-030-52200-1_9 | |
| Jul 20, 2023 at 17:01 | comment | added | Stopple | @Conrad Speiser's 1934 proof is topological, not analytic, and assumes implicitly that the level curves real (resp.) imaginary part of $\zeta(s)=0$ never pass through a zero of $\zeta^\prime(s)$, so the curves don't cross or branch. Spira showed unconditionally that RH=> the zeros of $\zeta^\prime$ lie to the right of the critical line, and Montgomery-Levin proved the other implication, also unconditionally. | |
| Jul 20, 2023 at 15:56 | comment | added | Conrad | I agree that looking at the paper cited above it seems so, but what is a bit weird is that Broughan's book Equivalents of RH Volume I cites same paper but notes that it doesn't quite imply the equivalence; I guess that you are right and indeed there is an equivalence and Broughan is just mistaken | |
| Jul 20, 2023 at 15:49 | comment | added | Agno | @Conrad If I interpret the answer to this MO-question correctly mathoverflow.net/questions/107938/…, the converse should be true as well. | |
| Jul 20, 2023 at 15:41 | comment | added | Conrad | I am not sure that RH is equivalent to the fact that the non-trivial zeros of ζ′(s) are on the right of the critical line, as I think it is only an implication (RH implies that but not the converse) | |
| Jul 20, 2023 at 15:35 | history | asked | Agno | CC BY-SA 4.0 |