= Character prime sums in the Chebotarev proof
{title2=$\sum_{\mathfrak p\,\mathrm{unramified}}\chi(g_{\mathfrak p})(N\mathfrak p)^{-s}=\langle\chi,1_G\rangle_G\log(1/(s-1))+O(1)$}
The logarithm from the <Euler product> of an <Artin L-function> has first-prime-power term $\sum\chi(g_{\mathfrak p})(N\mathfrak p)^{-s}$. Higher powers are uniformly bounded as real $s\downarrow1$: $|\chi(g)|\leq\chi(1)$ and $\sum_{\mathfrak p}\sum_{m\geq2}(N\mathfrak p)^{-ms}/m\leq2\sum_{\mathfrak p}(N\mathfrak p)^{-2}<\infty$. The finitely many ramified local factors also have bounded logarithms. The <Artin L-function order at one> therefore proves the displayed estimate. Expanding a conjugacy-class indicator by <character orthogonality> leaves only the trivial character's logarithmic term, with coefficient $|C|/|G|$. This proves <Dirichlet density> directly; a natural-density assertion needs additional analytic arguments.
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