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?

PS: I'm not looking for a patronizing lecture on what I should or should not know or be able to do. No one at my university does anything close to p-adic harmonic analysis (not even my advisor) so I have no one to turn to but the internet. Also, the vast preponderance of representation theory and algebraic number theory among the literature makes it very difficult for an analyst such as myself to read and make sense of. As such, I'm just looking for clarification so I can know the rules for working with these types of integral transforms so that I can move on with my research.

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