Timeline for Why are functional equations important?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 19, 2010 at 16:59 | comment | added | Kevin Buzzard | "do we care about the values of zeta (and other L-functions) to the left of the critical strip?" Absolutely! The values of the zeta function at negative integers are rationals, and hence make sense as p-adic numbers. Furthermore, for n negative, the map sending n to zeta(n) is p-adically continuous (for n in a fixed congruence class mod p-1) and so we can define a p-adic zeta function! p-adic L-functions have been used to e.g. prove Leopoldt's conjecture for abelian extensions of the rationals. However pi isn't a p-adic number and so this trick wouldn't work if we only restricted to +ve vals. | |
| Jan 15, 2010 at 16:57 | comment | added | Anweshi | Tate's thesis gives a nice explanation of the prime factors and the Gamma function in the functional equation for Dedekind zeta functions. We wouldn't have that nice explanation without a functional equation. | |
| Jan 15, 2010 at 16:07 | comment | added | Noah Snyder | This technique is originally due to Euler in his paper on the functional equation of the zeta function (note no notion of complex analysis yet!). | |
| Oct 23, 2009 at 15:20 | comment | added | engelbrekt | You can easily get the analytic continuation to the whole complex plane, strip by strip, by repeatedly integrating by parts in the representation obtained from the Euler-Maclaurin formula. | |
| Oct 23, 2009 at 12:31 | history | answered | David E Speyer | CC BY-SA 2.5 |