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 ? 3 Changed l to p. 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$B_k$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 $$B_k+\sum_{p-1|k}{1\over l}\in\mathbb{Z}p}\in\mathbb{Z}.$$ 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.