Before I ask the question, I need to recall what Bernoulli numbers $(B_k)_{k\in\mathbb{N}}$ are, and what von Staudt and Clausen discovered about them in 1840. The numbers $B_k$ B_k\in\mathbb{Q}$ are the coefficients in the formal power series $$ {T\over e^T-1}=\sum_{k\in\mathbb{N}}B_k{T^k\over k!} $$ so that $B_0=1$, $B_1=-1/2$, and it is easily seen that $B_k=0$ for $k>1$ odd.
Theorem (von Staudt-Clausen, 1840) Let $k>0$ be an even integer, and let $p$ run through the primes. Then
$$
B_k+\sum_{p-1|k}{1\over p}\in\mathbb{Z}.
$$
John Coates remarked at a recent workshop that the analogue of this theorem
for a totally real number field $F$ (other than $\mathbb{Q}$) is an open problem;
even a weak analogue would imply Leopoldt's conjecture for $F$. I missed the
opportunity of pressing him for details.
Question : What is the analogous statement over a totally real number field ?
