Timeline for Why does the Gamma-function complete the Riemann Zeta function?
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 18, 2019 at 2:15 | comment | added | The_Sympathizer | Note that this gamma formula does not complete the whole thing: in particular, you cannot derive the hallowed critical strip from it and the base sum-of-powers definition - which should make sense, as it suggests there is, in a sense, "too much complexity" and "I'm not gonna make it quiiite so eassie on youue" for the gamma function, alone, to capture. In that regard, perhaps, the gamma formula may not be as surprising as one may at first think. | |
| Apr 17, 2019 at 13:32 | answer | added | KConrad | timeline score: 18 | |
| Apr 2, 2017 at 16:33 | comment | added | Watson | For any even function $f$ belonging to the Schwartz space, we have $\widetilde f (s) \zeta(s) = \widetilde{\hat f}(1-s) \zeta(1-s)$, where $\widetilde g$ is the Mellin transform of $g$. Taking $f(y) = e^{-\pi y^2}$ yields the result. | |
| Oct 31, 2016 at 1:50 | answer | added | Anixx | timeline score: -2 | |
| Sep 10, 2014 at 16:31 | answer | added | paul garrett | timeline score: 22 | |
| Sep 10, 2014 at 5:39 | answer | added | Anixx | timeline score: 3 | |
| Jun 1, 2014 at 16:52 | comment | added | Lucian | They are both conceptually related to sums of powers. The $\zeta$ function itself is defined as a non-alternating sum of powers for $\Re(z)>1$, and as an alternating sum of powers (times a certain factor) for $\Re(x)\in(0,1)$ On the other hand, geometric shapes of the form $x^n+y^m=1$, called superellipses or Lame curves, are also bounded sums of powers. But by integrating $y=\sqrt[m]{1-x^n}$ or $x=\sqrt[n]{1-y^m}$ on $(0,1)$ we get the multiplicative inverse of the binomial coefficient ${m+n\choose n}={m+n\choose m}$, which is obviously expressible in terms of the $\Gamma$ function. | |
| Aug 2, 2010 at 3:07 | answer | added | Dr_Acula | timeline score: 14 | |
| Dec 4, 2009 at 4:44 | answer | added | David E Speyer | timeline score: 37 | |
| Dec 4, 2009 at 2:24 | vote | accept | Peter Arndt | ||
| Dec 3, 2009 at 22:11 | answer | added | Ricardo | timeline score: 10 | |
| Dec 3, 2009 at 14:27 | answer | added | Rob Harron | timeline score: 3 | |
| Dec 3, 2009 at 14:20 | comment | added | Harry Gindi | Have you ever read Emil Artin's monograph about the gamma function? | |
| Dec 3, 2009 at 13:43 | answer | added | Harald Hanche-Olsen | timeline score: 60 | |
| Dec 3, 2009 at 13:09 | answer | added | Leonid Positselski | timeline score: 60 | |
| Dec 3, 2009 at 12:16 | history | asked | Peter Arndt | CC BY-SA 2.5 |