$p$-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 $k\ge 1$?



What is the value of $p$-adic $\zeta$-function at positive integer point?
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](https://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions)), 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.


