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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]].\nLet $r_1$ be the number of real embeddings of $K$ and $r_2$ the number of pairs of complex conjugate non-real embeddings.\nLet $h_K$ be the [[class number|KTKK9B1X41]] of $K$, $R_K$ its [[regulator|RWK0HNF1NA]], $w_K$ the number of [[roots of unity|0YV2CKP2AV]] in $K$ and $d_K$ its [[discriminant|E0VZTNPJXA]].\nThen the [[Dedekind zeta function|KXS5DQDFYD]] $\\zeta_K(s)$ has a simple pole at $s = 1$, and\n$$\\lim_{s \\to 1^+} (s - 1)\\, \\zeta_K(s) = \\frac{2^{r_1} (2\\pi)^{r_2}\\, h_K\\, R_K}{w_K \\sqrt{\\abs{d_K}}}.$$\n\n## Remarks\n\nThe proof counts ideals of norm at most $x$ in each ideal class.\nThis count reduces to counting lattice points in a region of $\\R^{r_1} \\times \\C^{r_2}$ cut out using the units.\nSo its leading term is a volume; the volume brings in $R_K$ and $\\sqrt{\\abs{d_K}}$, and the classes bring in $h_K$.\nExample: for $K = \\Q$ one has $r_1 = 1$, $r_2 = 0$, $h = R = 1$, $w = 2$, $d = 1$, and the formula gives residue $1$, as for the Riemann zeta function.\nFor an imaginary quadratic field of discriminant $d \u003c 0$, the formula becomes $h_K = \\frac{w_K \\sqrt{\\abs{d}}}{2\\pi} L(1, \\chi_d)$, where $\\chi_d$ is the Kronecker character of $K$ and $L$ its [[Dirichlet L-function|9D2Z543EJQ]].\nFor $K = \\Q(i)$: $w = 4$, $d = -4$ and $L(1, \\chi_{-4}) = 1 - \\frac13 + \\frac15 - \\dots = \\frac{\\pi}{4}$, so $h_K = 1$.\n\n## Explanations\n\n### L1 human\n\nThe class number measures how far unique factorisation fails in a number field.\nThe formula says that this number, together with other basic invariants of the field, can be read off from the behaviour of an analytic function near the point $s = 1$.\nSo a purely algebraic quantity is computed by analysis.\n\n### L2 human\n\nFor an abelian field $K$, the function $\\zeta_K(s)$ is the product of $\\zeta(s)$ and Dirichlet $L$-functions, so the residue becomes a product of values $L(1, \\chi)$.\nThis gives Dirichlet's class number formulas for quadratic fields, and the fact that $L(1, \\chi) \\ne 0$ for real characters, which is the key step in [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].\nThe units enter through the regulator, which is defined by means of [[Dirichlet's unit theorem|1D99CSKX6N]].\n",
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