Timeline for Why does the Gamma-function complete the Riemann Zeta function?
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 5, 2017 at 21:27 | comment | added | Tom Copeland | Related: math.stackexchange.com/questions/143449/… and mathoverflow.net/questions/58004/… | |
| Oct 25, 2013 at 15:17 | comment | added | Harald Hanche-Olsen | @Aakumadula I think it's correct. As I wrote, some more work is needed to get the functional equation on the form that is commonly seen. The functional equation $\sqrt\pi\Gamma(2z)=2^{2z-1}\Gamma(z)\Gamma(z+\frac12)$ helps in this regard. | |
| Oct 25, 2013 at 14:35 | comment | added | Venkataramana | There is something wrong with this functional equation; the Gamma factor should be $\Gamma (s/2)$ and not $\Gamma (s)$. | |
| S Oct 25, 2013 at 12:43 | history | suggested | Daniel Miller | CC BY-SA 3.0 |
fixed TeX (there were newlines between the integrand and dt)
|
| Oct 25, 2013 at 12:35 | review | Suggested edits | |||
| S Oct 25, 2013 at 12:43 | |||||
| Dec 4, 2009 at 12:21 | comment | added | Harald Hanche-Olsen | @Ryan: That's okay. I learned a lot from these comments anyway, and that's worth more than a few points of rep on MO. | |
| Dec 4, 2009 at 5:04 | comment | added | Ryan Budney | Hi Harald, sorry, that was meant to be more of a sympathetic light-hearted joke. But understatement doesn't translate through ASCII. | |
| Dec 4, 2009 at 2:34 | comment | added | Peter Arndt | Thanks for the answer! Indeed I came to my question from the other side, being aware that there should be a factor for the infinite prime but not understanding why it involves the Gamma-function. This gives me an idea how Riemann got there. Checkmark goes to Leonid though, bringing me the good news that we live in the best of possible worlds :-) | |
| Dec 4, 2009 at 2:00 | comment | added | Rob Harron | Tate's thesis is rather well written. There's also the book of Ramakrishnan-Valenza "Fourier analysis on number fields" that develops the subject and goes over Tate's thesis. The title of the book is rather suggestive. Section 3.1 of Bump's book "Automorphic forms and representations" has an overview of Tate's thesis. | |
| Dec 4, 2009 at 1:48 | comment | added | Leonid Positselski | I guess Tate's dissertation has been published in "Algebraic Number Theory", edited by Cassels and Froelich. It must be well-written, but slightly complicated in that it deals with arbitrary number fields rather than just Q. For an introduction to p-adic numbers, I would suggest "P-adic numbers, p-adic analysis, and zeta-functions", by Neal Koblitz. | |
| Dec 4, 2009 at 1:40 | comment | added | Harald Hanche-Olsen | Again, thanks for the explanation. I should probably ask over at Leonid's question, but since this thread of comments is so long already – if I want to learn more, in order to understand Leonid's answer, but don't quite feel up to obtaining and reading John Tate's dissertation, is there a good place I can go? Bear in mind that I am not an algebraist, I just want to widen my horizon, not to understand every little detail in the argument. | |
| Dec 4, 2009 at 1:17 | comment | added | Leonid Positselski | @Harald: It means that you can embed the field Q of rational numbers into the p-adic field Q_p, for any prime number p, just as you can embed it into the field of real numbers R. The p-adic fields are just the other completions of Q that are there, in addition to R (completions with respect to other metrics). Given that, you can view the reals as another "prime number". This resembles adding the point at infinity to the affine line, to get the projective line. The "real prime" is that point at infinity. Formulas in number theory are supposed to include it on par with the ordinary primes. | |
| Dec 4, 2009 at 1:07 | comment | added | Rob Harron | @Harald: What Ben means about a real prime is that if you consider the set of equivalence classes of absolute values on Q there is one for each prime p, and the usual absolute value. This leads number theorists to consider the usual absolute value as an "infinite" prime. It is called real as it comes from the embedding of Q into R (whereas finite extensions of Q might embed into C, but not R, and hence have complex primes). The Riemann zeta function can be viewed as an Euler product of factors 1/(1-p^-s) and the gamma factor can be viewed as the factor coming from the infinite prime. | |
| Dec 4, 2009 at 0:17 | comment | added | Harald Hanche-Olsen | @Ben: Thanks for the explanation. Now I only have to figure out what “Q has a real prime” means. (As is probably clear by now, I'm an analysis guy, not an algebraist.) | |
| Dec 3, 2009 at 23:40 | comment | added | Rob Harron | @Ben Webster: though it is true that it is the real prime of Q that allows for the appearance of the gamma factor, the question was "Why the Gamma function?", not "Why is there another factor?". The real prime could be considered as a reason to have another factor, not a reason for that factor to be the gamma function. Harald's answer illustrates how the gamma function arises in the proof. | |
| Dec 3, 2009 at 23:18 | comment | added | Ben Webster♦ | Ryan- Actually yes. I think Tate understands the Riemann zeta function a lot better than Riemann ever did, though of course, that involved a lot of standing on the shoulders of giants (specifically, Tate knew about class field theory, and Riemann never had a chance). | |
| Dec 3, 2009 at 23:14 | comment | added | Ben Webster♦ | I was one of the downvotes. I'll say, I don't think your answer is wrong or problematic, I just think there's a much better answer, which hasn't been written properly: "Q has a real prime." I'm not familiar enough with the subject to write a good answer like that, but I'm familiar enough with the subject to say I don't think an answer which leaves it out should be at the top. | |
| Dec 3, 2009 at 23:01 | comment | added | Harald Hanche-Olsen | @Ryan: Now don't let us overreact. I am puzzled, not angry or disappointed. | |
| Dec 3, 2009 at 22:18 | comment | added | Ryan Budney | I guess Riemann's opinions about the Riemann zeta function aren't good enough for some people? | |
| Dec 3, 2009 at 15:24 | comment | added | Harald Hanche-Olsen | So far, this answer has received four upvotes and two downvotes. I am curious about the reason for the downvotes: I thought they were intended for off topic or wrong answers, especially the sort of answers you want to discourage, and I can't see that this answer is either. Perhaps I should have left the second half out of it, since it does not contribute much to the why question, but to me, that doesn't seem sufficient reason for a downvote. | |
| Dec 3, 2009 at 14:02 | history | edited | Harald Hanche-Olsen | CC BY-SA 2.5 |
Consistency of variable name
|
| Dec 3, 2009 at 13:43 | history | answered | Harald Hanche-Olsen | CC BY-SA 2.5 |