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My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I've been fam …

Let $S$ be a set of finitely many prime numbers. Then, define $\left|\cdot\right|_{S}:\mathbb{Q}\rightarrow\left[0,\infty\right)$ by: $$\left|x\right|_{S}\overset{\textrm{def}}{=}\prod_{p\in S}\left|x …

(This is a literature/reference question.) So... long story short: (1) In my present research, I needed a theory of continuous functions from the $p$-adic integers to the $q$-adic integers. Unable t …

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\ri …

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 …

Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to …