nt.number theory - Power series $f$ such that $f(zeta_{p^n}-1)=1/p^n$ for almost all $n geq 0$

As Brian pointed out, the principal parts problem always has a solution on the open unit disk, and Lazard's 1962 article "Les zéros des fonctions analytiques d’une variable sur un corps valué complet" gives a nice proof which is also rather explicit.

Your problem has additional symmetries so one can be a bit more explicit, as follows.

Let $H$ be the set of power series holomorphic on the open unit disk, let $varphi: H to H$ be the map defined by $varphi(f)(X)=f((1+X)^p-1)$ and let $d$ be defined by $d(f)(X)=(1+X)df/dX$. Note that $dvarphi=pvarphi d$. You can check that if you have a function $f$ such that $varphi(f)-pf = (1+X/2)log(1+X)/X$, then $f(0) neq 0$ and $f(zeta_{p^n}-1) = p^{-n} f(0)$ so this answers your problem.

Therefore you need to be able to solve an equation of the form $varphi(f)-pf = g$. By taking $d$ of both sides this gives $varphi(df)-df=dg/p$. Now you can solve an equation of the form $varphi(a)-a=b$ if $b in X cdot H$ by writing $a = sum_{n geq 0} varphi^n(b)$, which should converge in $H$ for its Fréchet topology. We have $d((1+X/2)log(1+X)/X) in X cdot H$ (this is what the $(1+X/2)$ was put in for), and once you know $df$, you get $f$ by integrating and adjusting the constant.