In the paper they prove that the series
$$
\zeta_{c, d, p}(s)=\sum_{n=0}^\infty f(n, p, c, d)p^{-ns}
$$
can be written as
$$
\zeta_{c, d, p}(s)=\sum_{N \triangleleft \widehat{(F_{c,d})_p}}\left|\widehat{(F_{c,d})_p}:N\right|^{-s}\left|\mathfrak G_p:\operatorname{Stab}_{\mathfrak G_p}\left(N\right)\right|^{-1}
$$
where $\widehat{(F_{c,d})_p}$ is the pro-$p$ completion of the free $p$-group of class $c$ on $d$ generators and $\mathfrak G_p=\operatorname{Aut}\widehat{(F_{c,d})_p}$.
It is an open problem whether the fractional part of $\left(\dfrac32\right)^n$ is dense in $[0...1]$.
The problem is: is $1$ a limit point of the above sequence?
An equivalent formulation is: $\forall \epsilon > 0: \exists n \in \Bbb N: 1 - \{1.5^n\} < \epsilon$ where $\{x\}$ denotes the fracti...
