The Artin conjecture states that for any Galois extension $E/K$ of global fields, and for any non-trivial simple character $\chi$ of $\textrm{Gal}(E/K)$. The Artin $L$-series $L(E/K,\chi,s)$ admits an analytic continuation holomorphic on $\mathbb{C}$.
A trivial consequence of inductive invariance is that:
$$
\zeta_E(s) = \zeta_K(s) \prod_{\chi \neq \chi_0}L(E/K,\chi,s)
$$
The product being taken over all non-trivial simple characters of $\textrm{Gal}(E/K)$.
A corollary of this is that for a Galois extension $E/K$, the ratio of the corresponding $\zeta$-functions, $\frac{\zeta_E(s)}{\zeta_K(s)}$ is holomorphic on $\mathbb{C}$. This has in fact been proved by other means – without assuming the Artin conjecture – and is known as the Aramata-Brauer theorem.
On p. 83 of $\textit{An Introduction to the Langlands Programme}$ Ehud de Shalit says that:
The corresponding theorem for non-Galois $K/k$ is open, although it would follow from the [Artin] conjecture.
That is to say, he claims that the holomorphicity of the ratio of zeta functions follows from the Artin conjecture.
I have been trying to prove this, but have had some difficulties circumventing the fact that we cannot make sense of the $L$-series related to non-Galois extensions $E/K$. One could take the normal closure (say $E \subset N$), but then one would be dealing with $N/K$, which would show that the ratio of the $\zeta$-functions corresponding to this extension is entire, but how could we relate this to $E/K$?
So why does the Artin conjecture imply the holomorphicity of the ratio of the zeta function?
Thank you for your attention.
Best Answer
If $L/K$ is a finite extension let $N/K$ be the Galois closure or any Galois extension containing $L$, $G=Gal(N/K),H=Gal(N/L),d=|G|/|H|=[N:L]$ then $$\zeta_L(s)=L(N/K,Reg_{G/H},s),\qquad \zeta_K(s)=L(N/K,1,s)$$
where $Reg_{ G/H}$ is the representation sending $g\in G$ to the matrix $\in GL_d(\Bbb{Z})$ of the permutation $xH\mapsto gxH$ of $G/H$,
and $1$ is the trivial representation $G\to \Bbb{C}^*$.
Thus $$\frac{\zeta_L(s)}{\zeta_K(s)}=L(N/K,\rho,s)$$ where $\rho$ is the representation such that $Reg_{G/H}\cong 1 \oplus \rho$, the Artin conjecture implies that the RHS is entire.
The Dedekind conjecture is that the LHS is entire.