Timeline for Why does LMFDB refer to L functions having coefficients of type $a_p-a_{p^2}$ instead of just $a_{p^2}$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 23, 2020 at 18:34 | comment | added | Milo Moses | @Wojowu ah, that makes more sense. Thank you for clarifying | |
| Nov 23, 2020 at 10:50 | comment | added | Wojowu | @MiloMoses That's true, I phrased myself poorly. Rather what I meant is that this is what allows the Euler factors to be contractible into a form you see for many common L-functions, namely $P(p^{-s})^{-1}$ for some polynomial $P$. | |
| Nov 23, 2020 at 6:25 | vote | accept | Milo Moses | ||
| Nov 23, 2020 at 5:47 | answer | added | Manipulator | timeline score: 1 | |
| Nov 23, 2020 at 3:30 | comment | added | Milo Moses | @Wojowu do you mean that these relations are what allows them to admit functional equations? Any Dirichlet series that is totally multiplicative should have an Euler product | |
| Nov 22, 2020 at 22:19 | comment | added | Kimball | See also: mathoverflow.net/a/287761/6518 | |
| Nov 22, 2020 at 21:21 | comment | added | Wojowu | Such relations between coefficients are precisely what enable the Dirichlet series to admit an Euler product, and near all L-functions have them. Perhaps the most famous of such is given by the Ramanujan conjectures on the $\tau$ function. | |
| Nov 22, 2020 at 20:39 | comment | added | Milo Moses | @Wojowu I assumed this wasn't the case since this implied too beautiful of a connection between $a_p$, $a_{p^2}$, and the higher order coefficients $a_{p^n}$. L functions never cease to amaze me. | |
| Nov 22, 2020 at 20:23 | comment | added | Milo Moses | Thank you! This makes a lot of sense. | |
| Nov 22, 2020 at 20:22 | comment | added | Wojowu | $a_n$ is the $n$-th coefficient of the Dirichlet series, i.e. we write the product as $\sum a_n/n^s$. With this notation, the coefficient in the product is indeed $a_p^2-a_{p^2}$. | |
| Nov 22, 2020 at 20:07 | history | asked | Milo Moses | CC BY-SA 4.0 |