= Solution
We prove the <Dirichlet density> form of the <Chebotarev density theorem>. Let $L/K$ be a finite <Galois extension> with <Galois group> $G$, let $C$ be a <conjugacy class> in $G$, and let $S_C$ consist of the <unramified> prime ideals of $K$ whose <Frobenius conjugacy class> is $C$. The assertion is
$$
\boxed{\delta(S_C)=\lim_{s\downarrow1}
\frac{\displaystyle\sum_{\mathfrak p\in S_C}(N\mathfrak p)^{-s}}
{\log(1/(s-1))}=\frac{|C|}{|G|}.}
$$
The same result holds for any union of <conjugacy classes>, by addition. The following proof uses exactly the allowed one-dimensional <Artin L-function> behavior at $1$ and the permitted induction and Euler-product properties.
First determine the <order of vanishing> at $1$ for an arbitrary <character of a representation> $\chi$ of $G$. The <Brauer induction> theorem supplies subgroups $H_j\leq G$, <linear characters> $\psi_j$ of $H_j$, and integers $n_j$, possibly negative, such that
$$
\chi=\sum_jn_j\operatorname{Ind}_{H_j}^G\psi_j.
$$
Put $K_j=L^{H_j}$. Compatibility of <Artin L-functions> with <induced representations> and direct sums gives, initially for $\operatorname{Re}s>1$,
$$
L_K(\chi,s)=\prod_jL_{K_j}(\psi_j,s)^{n_j}.
$$
Each $\psi_j$ is a one-dimensional <Galois representation> for $L/K_j$, so the stated analytic assumptions apply to every factor. A trivial $\psi_j$ contributes a simple pole; a nontrivial $\psi_j$ is holomorphic and nonzero at one. Thus the product is meromorphic near one and
$$
\operatorname{ord}_{s=1}L_K(\chi,s)
=-\sum_jn_j\langle\psi_j,1_{H_j}\rangle_{H_j}.
$$
The <character inner product> is $\langle\alpha,\beta\rangle_G=|G|^{-1}\sum_{g\in G}\alpha(g)\overline{\beta(g)}$. Applying <Frobenius reciprocity> to each induced term gives
$$
\boxed{\operatorname{ord}_{s=1}L_K(\chi,s)
=-\sum_jn_j\langle\operatorname{Ind}_{H_j}^G\psi_j,1_G\rangle_G
=-\langle\chi,1_G\rangle_G.}
$$
In particular, a nontrivial <irreducible character> has order zero, so its <Artin L-function> is holomorphic and nonzero near one. For the trivial character, $L_K(1_G,s)=\zeta_K(s)$, the <Dedekind zeta function> of the base field $K$, and its pole is simple. This argument establishes the behavior needed at one, not holomorphy everywhere for arbitrary <Artin L-functions>.
Next convert that order calculation into a prime sum. For an <irreducible character> $\chi$, choose a unitary representation of dimension $d_\chi$ affording it, and choose a <Frobenius automorphism> $g_{\mathfrak p}$ at each <unramified> prime. The logarithm specified by the absolutely convergent <Euler product> gives, for real $s>1$,
$$
\log L_K(\chi,s)
=\sum_{\mathfrak p\ \mathrm{unramified}}\sum_{m\geq1}
\frac{\chi(g_{\mathfrak p}^{\,m})}{m(N\mathfrak p)^{ms}}
+R_\chi(s).
$$
Only finitely many ramified primes occur. Their Euler-factor logarithms $R_\chi(s)$ stay bounded near one, since the relevant eigenvalues have modulus one and $(N\mathfrak p)^{-1}<1$. Also $|\chi(g)|\leq d_\chi$, so the contribution with $m\geq2$ is uniformly bounded for $s\geq1$ by
$$
\sum_{\mathfrak p}\sum_{m\geq2}
\frac{d_\chi}{m(N\mathfrak p)^{ms}}
\leq2d_\chi\sum_{\mathfrak p}(N\mathfrak p)^{-2}<\infty.
$$
Convergence follows, for example, from the convergent ideal series defining $\zeta_K(2)$. Therefore
$$
\sum_{\mathfrak p\ \mathrm{unramified}}
\frac{\chi(g_{\mathfrak p})}{(N\mathfrak p)^s}
=\log L_K(\chi,s)+O(1).
$$
Writing $a_\chi=\langle\chi,1_G\rangle_G$, the order computation means $L_K(\chi,s)=(s-1)^{-a_\chi}U_\chi(s)$, where $U_\chi$ is holomorphic and nonzero near one. Choose its local logarithm. Along the real interval approaching one, this and the Euler-product logarithm differ only by a fixed multiple of $2\pi i$. Consequently
$$
\boxed{\sum_{\mathfrak p\ \mathrm{unramified}}
\frac{\chi(g_{\mathfrak p})}{(N\mathfrak p)^s}
=a_\chi\log\frac1{s-1}+O(1).}
$$
Only the trivial <irreducible character> has $a_\chi=1$; all the others have $a_\chi=0$.
Finally, <character orthogonality> makes the <irreducible characters> an orthonormal basis of class functions. For $c\in C$, the coefficient of $\chi$ in the class indicator is $\langle1_C,\chi\rangle_G=(|C|/|G|)\overline{\chi(c)}$. Hence
$$
1_C(g)=\frac{|C|}{|G|}\sum_{\chi\in\operatorname{Irr}(G)}
\overline{\chi(c)}\chi(g).
$$
Sum this identity over the <unramified> primes with weight $(N\mathfrak p)^{-s}$. The finite character sum and the boxed estimates give
$$
\boxed{\sum_{\mathfrak p\in S_C}(N\mathfrak p)^{-s}
=\frac{|C|}{|G|}\log\frac1{s-1}+O(1).}
$$
Dividing by $\log(1/(s-1))$ proves the asserted <Dirichlet density>. The same Euler-product calculation with $\zeta_K$ gives $\sum_{\mathfrak p}(N\mathfrak p)^{-s}=\log(1/(s-1))+O(1)$, so using all primes as the denominator is equivalent. Finitely many ramified primes do not affect either limit. No prime-counting asymptotic or natural-density Tauberian step is required by this proof.
Back to article page
