Timeline for $\zeta(s+1)/\zeta(s)$
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 4, 2010 at 19:27 | comment | added | Jakob Katz | Also not quite what you're asking for, but for a different look at what I believe is the same relationship between Farey series and RH (not having read Franel's paper), check H.M. Edwards' book on the Riemann zeta function, paragraph 12.2. He references Franel, but proves the equivalence in question by other means. | |
| May 4, 2010 at 13:56 | vote | accept | Scott Guthery | ||
| May 4, 2010 at 13:00 | answer | added | Gerry Myerson | timeline score: 1 | |
| May 4, 2010 at 13:00 | comment | added | Gerry Myerson | So then all you want is a citation for Franel's paper? I'll write that in an answer, then. | |
| May 4, 2010 at 10:34 | comment | added | Scott Guthery | It is this equivalence that Franel uses in connecting Farey and Riemann. | |
| May 4, 2010 at 1:41 | answer | added | François G. Dorais | timeline score: 11 | |
| May 4, 2010 at 0:35 | comment | added | Gerry Myerson | The only Franel-Riemann connection I've been able to find concerns Farey series. How have you come to believe that Franel did what you say he did? | |
| May 3, 2010 at 22:14 | comment | added | Wadim Zudilin | The computation is discussed in G. P\'olya and G.~Szeg\"o, Problems and theorems in analysis, Vol.~II, Grundlehren Math. Wiss. 216, Springer-Verlag, Berlin et al. (1976), Division 8, Chapter 1, Sections 5--7. | |
| May 3, 2010 at 21:57 | comment | added | Mark Lewko | To compute $c(n)$ use the identities $1/zeta(s) = \sum_{n=1}^{\infty} \mu(n)/n^{s}$ (this follows from Mobius inversion) and $\zeta(s+1) = \sum_{n=1}^{\infty} 1/n^{s+1}$, which hold in an appropriate half-plane. | |
| May 3, 2010 at 21:49 | comment | added | muad | This is not the same series as you have mentioned but it may be useful none-the-less. Warren D. Smith, "Cruel and unusual behavior of the Riemann zeta function" secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/cruel.ps | |
| May 3, 2010 at 21:35 | history | asked | Scott Guthery | CC BY-SA 2.5 |