So, after scouring the entirety of the internet, I managed to find *one* (and, so far, *only* one) source that actually explains how to invert the $p$-adic mellin transform:

$$\mathscr{M}_{p}\left\{ f\right\} \left(s\right)\overset{\textrm{def}}{=}\frac{p}{p-1}\int_{\mathbb{Q}_{p}^{\times}}\left|\mathfrak{z}\right|_{p}^{s-1}f\left(\mathfrak{z}\right)d\mathfrak{z},\textrm{ }\forall s\in\mathbb{C}$$
where $d\mathfrak{z}$ is the haar probability measure on $\mathbb{Z}_{p}$, and where $s$ is a complex variable. The source in question are these notes from the University of Chicago, specifically, pages 72 and 73. However, being an analyst, the word "(un)ramified" gives me heart palpitations; I'll be honest, I don't know exactly how to interpret equations (4.15) and (4.16) from the notes (pages 72 & 73), nor their accompanying text. I know *just enough* to know that the integral I wrote above is what the writer meant in writing (4.14).

However, because of the maddening $t$ business in the notes—among other things—I cannot understand how to correctly write down the inversion formula, among other things. Before I ask my questions, let me just say:

i. I have no interest in integrating over anything other than complex-valued functions on $\mathbb{Z}_{p}$. For what I'm trying to learn, all the business about field extensions are needless complications in these notes that I'm trying to do away with as I explain the material to myself.

ii. I have no interest in Representation theory; I'm just an analyst whose work has led him into non-archimedean waters, and would like to know what the rules are for swimming in these circumstances.

Anyhow...

Is the correct way of writing (4.15):

$$\mathscr{M}_{p}^{-1}\left\{ F\right\} \left(\mathfrak{z}\right)=\frac{1}{2\pi i}\oint_{p^{-\sigma}\partial\mathbb{D}}\frac{F\left(s\right)}{\left|\mathfrak{z}\right|_{p}^{s}}ds$$ where $p^{-\sigma}\partial\mathbb{D}$ is the circle in $\mathbb{C}$ centered at $0$ of radius $p^{-\sigma}$, and where $\sigma$ is a positive real number.

Or is it: $$\mathscr{M}_{p}^{-1}\left\{ F\right\} \left(\mathfrak{z}\right)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{F\left(s\right)}{\left|\mathfrak{z}\right|_{p}^{s}}ds$$ where the contour is the line $\textrm{Re}\left(s\right)=\sigma$ in $\mathbb{C}$?

Or is it: $$\mathscr{M}_{p}^{-1}\left\{ F\right\} \left(\mathfrak{z}\right)=\frac{1}{2\pi i}\oint_{p^{-\sigma}\partial\mathbb{D}}\left|\mathfrak{z}\right|_{p}^{-s}F\left(-\frac{\ln s}{\ln p}\right)ds$$

Or is it something else, entirely?

Next, as a test-run, I tried to compute and then invert the transform of the constant $\mathbb{Z}_{p}$. Like in the notes, I computed: $$\mathscr{M}_{p}\left\{ \mathbf{1}_{\mathbb{Z}_{p}}\right\} \left(s\right)=\frac{1}{1-p^{-s}}$$ where $\mathbf{1}_{\mathbb{Z}_{p}}$ is the indicator function for $\mathbb{Z}_{p}$. This is the same as the notes, albeit they use $t=p^{-s}$ and write this as $\frac{1}{1-t}$.

However, when I try to use either of the above two attempts at interpreting the inversion formula (4.15), I end up with gobbledygook.

• The first formula I gave yields the constant function $$f\left(\mathfrak{z}\right)=\frac{1}{\ln p}$$

• The second formula yields (using the residue theorem): $$f\left(\mathfrak{z}\right)=\frac{1}{\ln p}\sum_{k\in\mathbb{Z}}\left|\mathfrak{z}\right|_{p}^{-\frac{2k\pi i}{\ln p}}=\frac{1}{\ln p}\sum_{k\in\mathbb{Z}}e^{2k\pi i\textrm{val}_{p}\left(\mathfrak{z}\right)}$$ which is always divergent.

• The third formula yields $f\left(\mathfrak{z}\right)=0$, because the integrand: $$\left|\mathfrak{z}\right|_{p}^{-s}F\left(-\frac{\ln s}{\ln p}\right)=\frac{\left|\mathfrak{z}\right|_{p}^{-s}}{1-p^{--\frac{\ln s}{\ln p}}}=\frac{\left|\mathfrak{z}\right|_{p}^{-s}}{1-s}$$ is holomorphic inside the unit disk.

None of these seem right to me, which makes me worry that none of the inversion formulae I've proposed are correct.

As such, I ask:

**(1)** What is the correct formula for the inversion of the $p$-adic mellin transform?

**(2)** What is the procedure for evaluating said integral? (Ex. Do I use the residue theorem, but ignore the existence of certain poles—if so, which ones?)

**(3)** More generally, given an $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ so that the integral: $$F\left(s\right)=\int_{\mathbb{Z}_{p}\backslash\left\{ 0\right\} }\left(f\left(\mathfrak{z}\right)\right)^{s}d\mathfrak{z}$$ exists and has an analytic continuation to a meromorphic or entire function of $s\in\mathbb{C}$, how would I go about inverting it to re-obtain $f$? What would be the inversion formula, are there any special cares I should take in computing it (ignoring certain singularities when computing residues, etc.)? And to what extent can I re-obtain $f$ in this way?

To anyone who has read this far: thank you very much for your time!