Timeline for Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2017 at 0:51 | comment | added | GH from MO | @Joël: If you want, remove your last two comments, and I will remove mine as well. | |
| Feb 13, 2017 at 18:01 | comment | added | GH from MO | @Joël: I don't see any contradiction. David Speyer shows that the Euler product for $\zeta(s)$ is divergent when $0<\Re(s)<1$. On the other hand, for a nontrivial Dirichlet character $\chi$, the Euler product of $L(s,\chi)$ converges in the half-plane $\Re(s)>1/2$ if and ony if the RH holds for $L(s,\chi)$. | |
| Jun 18, 2015 at 8:57 | vote | accept | Agno | ||
| Jun 18, 2015 at 7:43 | comment | added | GH from MO | @joro: Two related posts are mathoverflow.net/questions/63714/… and mathoverflow.net/questions/181241/… | |
| Jun 18, 2015 at 7:41 | comment | added | GH from MO | @joro: Roughly speaking, the reason is the oscillation of $\chi$ (when $\chi$ is nontrivial). More precisely, while $\sum p^{-s}$ over the primes clearly diverges unless $\Re s>1$, the oscillating sum $\sum\chi(p)p^{-s}$ has a chance to converge for smaller values of $\Re s$ assuming the primes are sufficiently well-distributed mod $n$ (where $n$ is the conductor of $\chi$). And, indeed, GRH implies that the latter sum converges for $\Re s>1/2$. | |
| Jun 18, 2015 at 7:03 | comment | added | joro | @GHfromMO Thank you. My mistake. So why the answer doesn't apply to the trivial character which is zeta AFAICT? | |
| Jun 18, 2015 at 6:37 | comment | added | GH from MO | @joro: The only character modulo $n$ that is nonnegative is the trivial character modulo $n$. | |
| Jun 18, 2015 at 6:36 | comment | added | joro | If $\chi$ is kronecker character it is nonnegative. Let $t=0$. Every term is real greater than one. I don't see how this will converge to the $L$ function (computing a single value above the correct is counterexample). | |
| Jun 17, 2015 at 21:40 | review | First posts | |||
| Jun 17, 2015 at 22:00 | |||||
| Jun 17, 2015 at 21:38 | history | answered | ABCDveve | CC BY-SA 3.0 |