Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 5 / b / Solution

Let $G=\mathrm{Cl}(K)$ and let $\hat{G}$ be its character group. Character orthogonality givesFor $\mathrm{Re}s>1$, the prime term of $\log L(s,\chi )$ isup to a function bounded as $s\to 1^{+}$, since higher prime powers converge there. The trivial character contributeswhereas every nontrivial character contributes $O(1)$ because its $L$-function is nonzero at $1$. Thereforeand the density is $\boxed{1/h_K}$.