In Coates' *$p$-adic L-functions and Iwasawa's theory*, the first of his congruence hypotheses is that $\delta\_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})\in \mathbb{Z}\_p$, where $\mathfrak{b},\mathfrak{c},\mathfrak{f}$ are ideals of the totally real base field $F$ with $\mathfrak{bc}$ prime to $\mathfrak{f}$ and $\mathfrak{c}$ prime to $p$, and
$$\delta\_n(\mathfrak{b},\mathfrak{c},\mathfrak{f}) := (\mathbb{N}\mathfrak{c})^{n+1}\zeta\_{\mathfrak{f}}(\mathfrak{b},-n)-\zeta\_{\mathfrak{f}}(\mathfrak{bc},-n)$$
is a difference of partial zeta functions. Later, Deligne-Ribet proved this en route to more general congruence results that encompass all of Coates' hypotheses, and in expositions of this ([e.g. here](https://math.berkeley.edu/%7Eribet/Articles/luminy.pdf)) they are phrased in terms of the similar-looking function defined as
$$\Delta\_{\mathfrak{c}}(-n,\chi) :=L(-n, \chi) - \chi(\mathfrak{c})(\mathbb{N}\mathfrak{c})^{n+1}L(-n, \chi),$$
and Coates' first congruence is restated as $\Delta\_{\mathfrak{c}}(-n,\chi) \in \mathbb{Z}\_p$ (or an extension thereof) where $\chi$ is a character of the ray class group mod $\mathfrak{f}$ and $\mathfrak{c}$ is an integral ideal prime to $p\mathfrak{f}$. I can't see how Coates' congruence follows from this. The other direction straightforward enough, since
$$\Delta\_{\mathfrak{c}}(-n, \chi) = -\sum\_{\mathfrak{b}} \chi(\mathfrak{bc}) \delta\_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})$$
where $\mathfrak{b}$ runs over any representatives of ray ideal classes mod $\mathfrak{f}$, but the finite Fourier inverted formula
$$\delta\_n(\mathfrak{b},\mathfrak{c},\mathfrak{f}) = -\frac{1}{|C\_\mathfrak{f}|}\sum\_{\chi} \overline{\chi}(\mathfrak{bc}) \Delta\_\mathfrak{c}(n,\chi)$$
where $\chi$ runs over $\widehat{C\_{\mathfrak{f}}}$ introduces this denominator, which could have factors of $p$. What am I missing here?



Why are the formulations of Deligne-Ribet/Coates congruences for L-functions equivalent?
I think you are misinterpreting Ribet's claim here.


In Ribet's article he defines $\Delta\_c(1 -k, \epsilon)$ for an **arbitrary function** $\epsilon: G\_f \to V$ where $V$ is a $\mathbf{Q}\_p$-vector space. It needn't be a group homomorphism into the unit group of a field extension of $\mathbf{Q}\_p$, even though this is by far the most familiar case. 


When Ribet restates Coates' congruences, he doesn't say precisely what he intends $\epsilon$ to be in congruence (A), but he certainly doesn't require that it should be a character; and the fact that it is assumed always to be $\mathbf{Z}\_p$-valued is a strong hint that it won't be a character in general (because $\mathbf{Z}\_p^\times$ contains too few roots of unity to be the target of interesting characters of most finite abelian groups). So it seems clear that he intends $\epsilon$ to be an arbitrary $\mathbf{Z}\_p$-valued function.


So you can recover Coates' formulation of the first congruence from Ribet's formulation just by setting $\epsilon$ to be the indicator function of a specific element of $G\_f$.


