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$p$-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 $k\ge 1$?

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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.

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  • $\begingroup$ Thanks! Do I understand correctly that "residue" $\lim_{n\to\infty}B_{2ap^n}$ at the pole $1$ can not be defined correctly? $\endgroup$ – Alexey Ustinov Oct 22 '15 at 3:10
  • $\begingroup$ If $p \neq 2$, then the negative even integers are dense in $\mathbb{Z}_p$; continuity would imply $\zeta_p$ is identically zero.if it were zero at the even integers. $\endgroup$ – user13113 Oct 22 '15 at 5:33
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    $\begingroup$ @Hurkyl, The zeta function is not a continuous function on $\mathbf{Z}_p$; in Koblitz' construction, for p > 2 it is a collection of (p-1) continuous functions (one for each residue class modulo p-1). Since p-1 is even your argument does not work. (A more high-powered approach is to view $\zeta_p$ as a function on the space $Hom(\mathbf{Z}_p^\times, \mathbf{C}_p^\times)$, which is a disjoint union of p-1 copies of the open unit disc). $\endgroup$ – David Loeffler Oct 22 '15 at 6:34
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    $\begingroup$ @AlexeyUstinov The residue at the pole is very easy to define if you set things up correctly; it is 1 - 1/p. This is explained very nicely in one of the earlier chapters of Washington's book on cyclotomic fields. $\endgroup$ – David Loeffler Oct 22 '15 at 6:36
  • $\begingroup$ Yes, I have found the residue in H. Cohen's book (Proposition 11.3.9). $\endgroup$ – Alexey Ustinov Oct 22 '15 at 6:41
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I agree nothing much is known, but there are a number of formulas linked to the values at positive integers of p-adic L-functions: see Section 11.3 of my GTM 240 book (sorry for the self-advertisement).

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    $\begingroup$ Welcome to MO, Professor Cohen! $\endgroup$ – Todd Trimble Oct 22 '15 at 3:12
  • $\begingroup$ The relation between $\zeta_{p}(k)$ and $\zeta(k)$ is also described in section 4.3.3 of Fonctions $L$ $p$-adiques des représentations $p$-adiques (Astérisque 229) $\endgroup$ – Olivier Oct 22 '15 at 11:15
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Let me mention some irrationality results about p-adic zeta values:

In 2005, Frank Calegari arxiv link proved that for $p=2,3$, the p-adic zeta values $\zeta_p(3)$ is irrational (hence nonzero). It can be viewed as the p-adic analogue of Apéry's theorem. Later, Frits Beukers arxiv link gave an alternating proof of Calegari's results (and the irrationality for some other p-adic L-values).

In 2010, Pierre Bel arxiv link obtained some partial results on the irrationality of p-adic Hurwitz zeta values. See also his 2018 paper on $\zeta_p(4,x)$.

Very recently (in 2020), Johannes Sprang arxiv link showed that $$ \dim_{K}(K+\zeta_p(2)K+\zeta_p(3)K+\cdots+\zeta_p(s)K) \geqslant \frac{(1-o(1))\log s}{2[K:\mathbb{Q}](1+\log 2)}, $$ which is the analogue of Ball-Rivoal theorem for p-adic zeta values.

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