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In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^{s-1}f\left(x\right)dx$$ in closed form computing the inverse mellin transform using the Residue theorem gets us formulae (usually asymptotic, but sometimes exact) for the behavior of $f\left(x\right)$ as $x$ decreases to $0$ and/or as $x$ increases to $\infty$. Ex:

$$\int_{0}^{\infty}x^{s-1}\sum_{n=0}^{\infty}e^{-2^{n}x}dx=\frac{\Gamma\left(s\right)}{1-2^{-s}}$$ implies: $$\sum_{n=0}^{\infty}e^{-2^{n}x}=\frac{1}{2}-\frac{\gamma+\ln x}{\ln2}+\frac{1}{\ln2}\sum_{k\in\mathbb{Z}^{\times}}\Gamma\left(\frac{2k\pi i}{\ln2}\right)x^{-\frac{2k\pi i}{\ln2}}-\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{2^{n}-1}\frac{x^{n}}{n!}$$ an (emotionally satisfying!) identity which holds for all x>0.

But, now consider the p-adic analogue: $$\mathscr{M}_{p}\left\{ f\right\} \left(s\right)=\int_{\mathbb{Z}_{p}}\left|\mathfrak{y}\right|_{p}^{s-1}f\left(\mathfrak{y}\right)d\mathfrak{y}$$ where $d\mathfrak{y}$ is the haar probability measure on $\mathbb{Z}_{p}$ and $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ is sufficiently nice. Is there a similar interpretation for the inverse mellin transform of $\mathscr{M}_{p}\left\{ f\right\} $ in terms of the asymptotics of $f$? Answers or references would be appreciated.

Addendum: I'm asking this because I have specific p-adic mellin transforms whose analytic continuations I've computed, and I would like to know what I can do with them, and how—if at all—I can use them to make conclusions about the asymptotics or value distributions of the functions I transformed.

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Yes, definitely! This is one of the key ideas underlying Tate's thesis, and the whole theory of local L-factors for smooth representations of p-adic groups that grew out of it. Try reading Bump's book on automorphic forms for an introduction.

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  • $\begingroup$ Is there a exposition of the analytical aspects of this p-adic mellin transform that isn't done as part of broader program of representation theory or the like? I'm asking because I'm working with such transforms in a context that has absolutely nothing to do with adeles, representations, or the like—just an analytical problem over the 2-adic integers. I've read robert's book on p-adic analysis, but it focuses on p-adic valued functions, rather than real and complex valued functions on the p-adics. $\endgroup$ – MCS Feb 8 at 3:34

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