Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer: $$ S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k $$ Convergence is easy to see. This can also be written as $$ S_{p,k} = \frac{1}{2} \sum_{0\neq x \in B_p} x^{-k} $$ where $$ \begin{aligned} B_p &:= \big\{\frac{a}{p^r}\; : \; a\in\mathbb{Z},\; r\in\mathbb{N},\; |a|_{\infty} \leq p^r\big\}\\ &=\{x\in\mathbb{Q} : |x|_v \leq 1 \text{ at all places } v\neq p\}\\ \end{aligned} $$ (Here, $|\cdot|_\infty$ is the usual real absolute value, and “places” [of $\mathbb{Q}$] mean either a prime $\ell$ or the symbol $\infty$.) The point of this last presentation is that it suggests the analogy with $B_\infty = \mathbb{Z}$, where $S_{\infty,k} = \sum_{n=1}^{+\infty} n^{-k}$ is the usual Riemann zeta function (at even integers): in a certain sense, the $S_{p,k}$ are a $p$-adic analogue of the $\zeta(k)$ (though probably not the most intelligent or satisfactory analogue).
Question(s): Have these $S_{p,k}$ appeared in the literature? Do they have a name? Do they satisfy some known relations? (E.g., can $S_{p,4}$ be expressed in function of $S_{p,2}^2$? Experimentally, the ratio $S_{p,4}/S_{p,2}^2$ does not appear to be rational, but maybe something else can be said.)
For what it's worth, Wolstenholme's theorem $\sum_{a=1}^{p-1}\frac{1}{a^2}\equiv 0\pmod{p}$ tells us that $S_{p,2} \equiv 1\pmod{p^4}$. There seem to be many generalizations in various directions (see, e.g., here) involving Bernoulli numbers, Kummer congruences, or the Kubota-Leopoldt $p$-adic zeta function, but I couldn't find a direct connection with the above quantity.
