If $\chi$ is even, why is $B_{n,\chi\omega^{-n}}$ nonzero?

Question

In Washington's book on cyclotomic fields, he mentions on page 57 in my edition (in the remarks after Theorem 5.11) that

If $\chi$ is an even Dirichlet character then $B_{n,\chi\omega^{-n}}\neq0$,

where $B_{n,\psi}$ are the generalized Bernoulli numbers and $\omega$ is the Teichmuller character. I feel like this must just following from some property of the generating function $$ \sum_{a=1}^f\frac{\psi(a)te^{at}}{e^{ft}-1}=\sum_{n=0}^\infty B_{n,\psi}\frac{t^n}{n!}, $$ where $f$ is the conductor of $\psi$, but I can't seem to prove it. I know that the generating function has the same parity as $\psi$, but this seems only to be helpful in showing that certain Bernoulli numbers vanish. Is there some property I'm missing?

P.S. Although this is a question purely about Bernoulli numbers, I've included a few extra tags just because I think people who have read Washington's book probably have my answer. Feel free to delete them if need be.

Answer 1

• For $p$ odd prime let $\omega$ be a Dirichlet character modulo $p$ (completely multiplicative $p$-periodic function $\Bbb{Z\to C}$) of order $p-1$,

  For $\chi$ a Dirichlet character and integer $n$, $\Phi(a)=\chi(a)\overline{\omega}(a)^n$ is a Dirichlet character, let $\phi$ be the primitive Dirichlet character modulo $d$ underlying it.

• For $n$ even and $\chi$ even (as function on $\Bbb{Z}$) then $\phi$ is even, its Dirichlet L-function has the functional equation $$\Lambda(s,\phi)= d^{-s/2} \pi^{-s/2}\Gamma(s/2)L(s,\phi) = \xi_\phi \Lambda(1-s,\overline{\phi})$$ Since the RHS has an Euler product, for odd positive integer $k$, $L(-k,\phi)\ne 0$.

• On the other hand the analyticity for $\Re(s)> -M$ of $$\Gamma(s)L(s,\phi) -\sum_{m=0}^M \frac{B_{m,\phi}}{m!(s+m-1)}=\int_0^\infty (x\sum_{a=1}^\infty \phi(a)e^{-ax}-1_{x < 1}\sum_{m=0}^M \frac{B_{m,\phi}}{m!}x^m)x^{s-2}dx$$ implies $$L(-k,\phi)=(-1)^k\frac{B_{k+1,\phi}}{k+1}\ne 0$$

• It works the same way when $\phi $ is odd except that $$\Lambda(s,\phi)= d^{-(s+1)/2} \pi^{-(s+1)/2}\Gamma((s+1)/2)L(s,\phi) = \xi_\phi \Lambda(1-s,\overline{\phi})$$ means for even positive integer $k$, $L(-k,\phi)\ne 0$ (it stays true for $k=0$ but we also need the non-vanishing of $L(1,\phi)$)