There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition [here](https://math.berkeley.edu/~ribet/Articles/luminy.pdf), more precisely see Theorem 2.1.


What is confusing me is that they don't seem to use the L-functions with the Euler factor at p removed (which I will call the p-adic L function) but I thought that the Kummer congruences only held for the p-adic L functions. For instance, for the case of Dirichlet characters, I have only seen the theorem stated for p-adic L functions, for instance see Theorem 2.2 [here](http://math.stanford.edu/~conrad/DarmonCM/2011Notes/dirichlet.pdf). See also the [wiki page](https://en.wikipedia.org/wiki/Kummer%27s_congruence) for Kummer congruences without a character but with a higher order congruence. 


Conceptually, I thought of the Kummer congruences as equivalent to the continuity of the p-adic L function but I don't believe the L-function without the p Euler factor removed is continuous.


What am I missing? Is it true that the Kummer congruences (even in the case of just Dirichlet characters) or am I misinterpreting the Deligne-Ribet result?



Kummer congruences for totally real number fields
I think that the point lies in the difference between a *primitive* and *imprimitive* $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as the one found in Washington's *Introduction to Cyclotomic fields*, **Theorem 5.11**) gives the interpolation
$$
L\_p(1-n,\chi)=(1-\chi\omega^{-n}(p)p^{n-1})L(1-n,\chi)\qquad n\geq 1
$$
and this is shown to be equivalent to the Kummer congruences that you quote, namely
$$
(1-\chi\omega^{-n}(p)p^{n-1})\frac{B\_{\chi\omega^{-n},n}}{n}\equiv (1-\chi\omega^{-m}(p)p^{m-1})\frac{B\_{\chi\omega^{-m},m}}{m}\pmod{p^a}
$$
whenever $m\equiv n\pmod{p^{a-1}}$ and $\chi=\omega^r$ is a power of the Teichmüller character. The value $\chi\omega^{-n}(p)=\omega^{r-n}(p)$ is considered, in both statements, as the value of the *primitive* character attached to $\omega^{r-n}$ at $p$, in the following sense (the discussion is taken from Chapter 3 of Washington's book). When $r\not\equiv n\pmod{p-1}$, the character $\omega^{r-n}$ has conductor $p$ and is primitive; the value $\omega^{r-n}(p)$ is $0$, hence the "correction factor" disappears. The interesting part arises when $r\equiv n\pmod{p-1}$, so that $\omega^{r-n}$ is the trivial character of conductor $p$: this is an imprimitive character, whose primitivisation is the trivial character of conductor $1$, whose value at $p$ is $1$: the "correction factor" is there and takes the value $(1-p^{n-1})$. This, as Washington remarks right after his **Theorem 5.11** is the Euler factor at $p$ of the complex $L$-function defined on page 31 of his book ($2^\text{nd}$ edition), where the product runs over *all* primes.


In Deligne--Ribet the situation is different, and so is in Ribet's report that you quote. Indeed, in equation $(1.1)$ of his report he defines the $L$-function of a character $\varepsilon$ on a ray class group $G\_\mathfrak{f}$ as a Dirichlet series over all prime-to-$\mathfrak{f}$ integral ideals. In section 4 of the report, he applies this to the limit $G\_{\mathfrak{f}\mathfrak{p}^\infty}$. To make the bridge with Washington's situation, take $K=\mathbb{Q}$ as base field, and $\mathfrak{f}=1$. As character, chose $\varepsilon=\omega^{r-n}$ with $r\equiv n\pmod{p-1}$ (since otherwise, as discussed, Washington's and Ribet's expressions coincide). Then the $L$-function considered by Ribet is given by
$$
L(s,\omega^{r-n})=\prod\_{\ell\neq p}(1-\omega^{r-n}(\ell)\ell^{-s})^{-1}\qquad\operatorname{Re}(s)>1
$$
because $\omega^{r-n}$ is seen as character on $G=G\_{p^\infty}$ of conductor $p$, and the sum defining $L(s,\omega^{r-n})$ is performed only over primes $\ell\neq p$. The corresponding Kummer congruences get modified accordingly.


It might be helpful, to convince yourself of this discussion, to compare the interpolation formula proposed by Washington in **Theorem 5.11** and the one proposed by Ribet in his final section on page 13, in the paragraph preceeding equation $(4.6)$: he claims that $L\_p(1-k,\varepsilon)=L(1-k,\varepsilon\omega^{-k})$, showing the difference with Washington's.


