For a totally real field F, the Deligne-Ribet p-adic L-function provides a generalization of the p-adic Riemann zeta function exists, namely the p-adic Dedekind zeta function $\zeta_{F,p}$. \zeta_{F,p}$(as proved independently by Deligne–Ribet (Inv Math 59), Cassou-Noguès (Inv Math 51), and Barsky (1978)). One link between these and the Leopoldt conjecture is through the p-adic analytic class number formula which is the main theorem of Colmez's "Résidue en s = 1 des fonctions zêta p-adiques":adiques" (Inv Math 91): Added (2010/04/09): So here's how you can use von Staudt–Clausen to see that the$p$-adic zeta function (of Q) has a pole at s = 1. It is clear from your statment of vS–C that it is saying that for$k\equiv0\text{ (mod }p-1)$,$B_k\equiv -1/p\text{ (mod }\mathbf{Z}_p)$(i.e. it is not$p$-integral). Let$k_i=(p-1)p^i$, the$k_i$is$p$-adically converging to 0, so$\zeta_p(1-k_i)$is approaching$\zeta_p(1)$(since$\zeta_p(s)$is$p$-adically continuous, at least for$s\neq1$). By the aforementioned interpolation property of$\zeta_p(1-k)$, we have$$v_p(\zeta_p(1-k_i))=v_p(B_{k_i}/k_i)=-1-i\rightarrow -\infty$$hence$1/\zeta_p(1-k_i)$is approaching 0. 3 clean up I also can't answer the question, but I'll say some things that could help. One thing von Staudt-Clausen tells you is the denominator of the Bernoulli number$B_k$when p-1|k: B_k$: it is precisely, the product of primes p for which p-1|k $p-1\mid k$ (when $p-1\nmid k$, a result of Kummer says that $B_k/k$ is p-integral). As Buzzard commented, the Bernoulli numbers should be thought of (at least in this situation) as appearing in special values of p-adic L-functions, p-adic L-functions, specifically, for k a positive integer $$\zeta_p(1-k)=(1-p^{k-1})(-B_k/k),$$ where $\zeta_p$ is the p-adic Riemann zeta function (see chapter II of Koblitz's "p-adic numbers, p-adic analysis, and zeta-functions", for example). For a totally real field F, the Deligne-Ribet p-adic L-function p-adic L-function provides a generalization of the p-adic p-adic Riemann zeta function, namely the p-adic p-adic Dedekind zeta function $\zeta_{F,p}$. One link between these and the Leopoldt conjecture is through the p-adic p-adic analytic class number formula which is the main theorem of Colmez's "Résidue en s=1 s = 1 des fonctions zêta p-adiques": p-adiques": $$\lim_{s\rightarrow1}(s-1)\zeta_{F,p}(s)=\frac{2^{[F:\mathbf{Q}]}R_phE_p}{w\sqrt{D}}$$ where h is the class number, $E_p$ $E_p=\prod_{\mathfrak{p}\mid p}\left(1-\mathcal{N}(\mathfrak{p})^{-1}\right)$$is a product of Euler Euler-like factors, w = 2 is the number of roots of unity, D is the discriminant and R_p is the interesting part here: the p-adic p-adic regulator (as Colmez notes, \sqrt{D} and R_p both depend on a choice of sign, but their ratio does not). Theorem: The Leopoldt conjecture is equivalent to the non-vanishing of the p-adic p-adic regulator. (For this, see, for example, chapter X of Neukirch-Schmidt-Wingberg's "Cohomology of number fields"). A clear consequence of this is that if \zeta_{F,p} does not have a pole at s=1s = 1, then the Leopoldt conjecture is false for (F,p). F, p). Perhaps an understanding of the denominators of values of \zeta_{F,p} could lead to an understanding of the pole at s=1 s = 1 of \zeta_{F,p}. 2 Fixed typo in first displayed equation ("B_k/b") I also can't answer the question, but I'll say some things that could help. One thing von Staudt-Clausen tells you is the denominator of the Bernoulli number B_k when p-1|k: it is precisely, the product of primes p for which p-1|k (when p-1\nmid k, a result of Kummer says that B_k/k is p-integral). As Buzzard commented, the Bernoulli numbers should be thought of (at least in this situation) as appearing in special values of p-adic L-functions, specifically, for k a positive integer$$\zeta_p(1-k)=(1-p^{k-1})(-B_k/b),$$\zeta_p(1-k)=(1-p^{k-1})(-B_k/k),$$ where$\zeta_p$is the p-adic Riemann zeta function (see chapter II of Koblitz's "p-adic numbers, p-adic analysis, and zeta-functions", for example). For a totally real field F, the Deligne-Ribet p-adic L-function provides a generalization of the p-adic Riemann zeta function, namely the p-adic Dedekind zeta function$\zeta_{F,p}$. One link between these and the Leopoldt conjecture is through the p-adic analytic class number formula which is the main theorem of Colmez's "Résidue en s=1 des fonctions zêta p-adiques": $$\lim_{s\rightarrow1}(s-1)\zeta_{F,p}(s)=\frac{2^{[F:\mathbf{Q}]}R_phE_p}{w\sqrt{D}}$$ where h is the class number,$E_p$is a product of Euler factors, w is the number of roots of unity, D is the discriminant and$R_p$is the interesting part here: the p-adic regulator. Theorem: The Leopoldt conjecture is equivalent to the non-vanishing of the p-adic regulator. A clear consequence of this is that if$\zeta_{F,p}$does not have a pole at s=1, then the Leopoldt conjecture is false for (F,p). Perhaps an understanding of the denominators of values of$\zeta_{F,p}$could lead to an understanding of the pole at s=1 of$\zeta_{F,p}\$.