Timeline for The formula for (and computation of) the inverse p-adic mellin transform
Current License: CC BY-SA 4.0
14 events
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| Feb 9, 2020 at 23:40 | history | edited | KConrad | CC BY-SA 4.0 |
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| Feb 9, 2020 at 23:29 | comment | added | MCS | Let us continue this discussion in chat. | |
| Feb 9, 2020 at 6:00 | comment | added | KConrad | The difference between Prop. 4.6 and Prop. 4.7 is that the integral in Prop. 4.6 holds for all $\sigma \in \mathbf R$ while the integral in Prop. 4.7 holds for $\sigma > 0$. | |
| Feb 9, 2020 at 5:59 | comment | added | KConrad | Please don't use the exact same notation for equation references and theorem/proposition references. Your "(4.14)" means equation 4.14, but your similar notation "(4.6)" and "(4.7)" mean Prop. 4.6 and Prop. 4.7, not equations 4.6 and 4.7. | |
| Feb 9, 2020 at 5:51 | comment | added | MCS | Also, if the constant function on the integer ring of the field isn't a good candidate for (4.6), why does (4.7) then appear to immediately proceed to apply (4.14) to said function? | |
| Feb 9, 2020 at 5:48 | history | edited | KConrad | CC BY-SA 4.0 |
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| Feb 9, 2020 at 5:38 | comment | added | MCS | I don't know how to simplify it further without an explicit formula for the $\eta$. I'm also unsure of how to get rid of the $\times$ in the $\mathbb{Z}_{p}^{\times}$, or make the $\mathbb{Z}_{p}$-unit $\frac{a}{p^{m}}$ into a 1. | |
| Feb 9, 2020 at 5:30 | comment | added | MCS | So far, what I've got for the integral is: $\int_{\mathbb{Q}_{p}^{\times}}\left[\mathfrak{z}\overset{p^{n}}{\equiv}a\right]\left|\mathfrak{z}\right|_{p}^{s-1}\eta\left(\mathfrak{z}\right)d\mathfrak{z}=\frac{1}{p^{ms}}\int_{\frac{a}{p^{m}}+p^{n-m}\mathbb{Z}_{p}^{\times}}\left|\mathfrak{z}\right|_{p}^{s-1}\eta\left(p^{m}\mathfrak{z}\right)d\mathfrak{z}$ where $\left|a\right|_{p}=p^{-m}>p^{-n}$, and where the $\eta$ is the one you mentioned above. $\left[\mathfrak{z}\overset{p^{n}}{\equiv}a\right]$ is the iverson bracket for $a+p^{n}\mathbb{Z}_{p}$ (1 if true, 0 if false). | |
| Feb 9, 2020 at 4:45 | comment | added | KConrad | I am using $\xi_A$ in my post for the characteristic function of a set $A$, so please don't use $\xi$ in a comment to mean a character on $\mathbf Z_p$ (or $\mathbf Q_p$). | |
| Feb 9, 2020 at 4:31 | comment | added | MCS | So far, I've got $\chi\left(x\right)=\left|x\right|_{p}^{s}\times?$ as the formula for the $\chi$ in the integral. Need to know what the ? is. | |
| Feb 9, 2020 at 4:21 | comment | added | MCS | Some questions: (1) does any of this simplify (even slightly) if we assume that the functions we are transforming are defined only on $\mathbb{Z}_{p}$ (if so, how)? (2) How can we write a generic $\eta$ here "explicitly"? (Ex: like how, for a character $\xi$ on $\mathbb{Z}_{p}$, the "explicit" form of $\xi$ is $\xi\left(\mathfrak{z}\right)=e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}$ for some $t\in\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}$, treated here as a rational number in $\left[0,1\right)$). I can't know if I can do the integral or not until I see it written explicitly. xD | |
| Feb 9, 2020 at 3:58 | comment | added | MCS | Woo! Thanks so much! :D | |
| Feb 9, 2020 at 3:57 | vote | accept | MCS | ||
| Feb 9, 2020 at 4:22 | |||||
| Feb 9, 2020 at 3:18 | history | answered | KConrad | CC BY-SA 4.0 |