[Math] the value of $p$-adic $\zeta$-function at positive integer point

Question

$p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz) or, more precisely, it's corrected values
$$\zeta_p(1−k):=-(1-p^{k-1})\frac{B_k}{k}.$$
What is known about the values $\zeta_p(k)$ for $k\ge 1$?

Answer 1

If you use this definition, then $\zeta_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers.

What about the odd integers? At $k = 1$ there is a pole, unsurprisingly. At odd $k \ge 3$ the value is extremely mysterious, just as the complex zeta values $\zeta(k)$ are. There is an interpretation of the odd $p$-adic zeta values in terms of a $p$-adic regulator map in $K$-theory (see this question), but this is tough to get explicit information out of.

As an example of how little we understand these numbers, I believe it's an open problem whether the values $\zeta_p(k)$ for odd $k \ge 3$ are always non-zero, although this is certainly expected.