Timeline for Historical question in analytic number theory
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2011 at 18:31 | comment | added | paul garrett | I had the impression that Dedekind knew how to meromorphically continue just a teensy bit, even before Landau, but I may be mistaken. | |
| Jan 30, 2010 at 23:54 | vote | accept | David Hansen | ||
| Jan 29, 2010 at 21:01 | answer | added | KConrad | timeline score: 27 | |
| Jan 29, 2010 at 15:15 | answer | added | mql | timeline score: 3 | |
| Jan 29, 2010 at 7:23 | answer | added | maks | timeline score: 3 | |
| Jan 28, 2010 at 5:24 | comment | added | KConrad | Dedekind did not get analytic continuation of "his" zeta-functions beyond Re(s) > 1. The first person to prove analytic continuation of Dedekind zeta-functions (in general) a bit to the left of Re(s) = 1 was Landau, in 1903 I believe. He got continuation as far as Re(s) > 1 - 1/[K:Q], where K is the number field whose zeta-function you're dealing with. This is treated in Lang's Algebraic Number Theory. Before Landau, the density business involved limits as s approaches 1 from the right, as David writes. That one-sided limit does not imply anything about complex-analyticity around s = 1. | |
| Jan 28, 2010 at 2:19 | answer | added | KConrad | timeline score: 27 | |
| Jan 27, 2010 at 15:11 | answer | added | Matt Young | timeline score: 11 | |
| Jan 27, 2010 at 14:56 | comment | added | Anweshi | I mean the Dedekind zeta function of the cyclotomic field, which is a product of the Dirichlet $L$-functions for various characters of its Galois group which is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$, has a simple pole at $s = 1$. This is a significant step in the proof of Dirichlet's theorem. The answer to your question could be Dedekind, since the name occurs to me, and the zeta function of the number fields are named after him. | |
| Jan 27, 2010 at 14:48 | comment | added | David Hansen | I think it was earlier than Hecke, but my knowledge of the 19th century literature is very poor. Dirichlet's L-functions do not have any poles. You mean that the zeta function has a pole? Euler and everyone after him knew that it diverged like 1/(s-1) as s--->1 for s real, which is weaker than knowing anything about a pole, but sufficient for some applications ("Dirichlet density" of primes in arithmetic progressions). | |
| Jan 27, 2010 at 14:45 | comment | added | Anweshi | I would imagine it is Hecke. But I think Dirichlet himself did make use of the fact that the L-function has a simple pole at $s=1$. So in a sense he was complex analytic. | |
| Jan 27, 2010 at 14:42 | history | asked | David Hansen | CC BY-SA 2.5 |