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  "body": "\n## Statement\n\nLet $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\\C$ that is not constant, with [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f)$.\nLet $q \\ge 3$, and let $a_1, \\dots, a_q$ be distinct points of the [[Riemann sphere|0CDMXGPSES]] $\\C \\cup \\set{\\infty}$.\nFor $a$ in the Riemann sphere, let $\\overline{N}(r, a)$ be the counting function of the solutions of $f(z) = a$ counted without multiplicity, as in [[Nevanlinna theory|1EHH63VNAD]].\nThen there is a set $E \\subseteq (0, \\infty)$ of finite Lebesgue measure such that\n$$(q - 2)\\, T(r, f) \\le \\sum_{j=1}^{q} \\overline{N}(r, a_j) + S(r, f) \\qquad (r \\notin E),$$\nwhere $S(r, f) = O\\big(\\log T(r, f) + \\log r\\big)$ as $r \\to \\infty$ outside $E$.\n\n## Remarks\n\nThe proof rests on the lemma on the logarithmic derivative, which says that the mean of $\\log^+ \\abs{f'/f}$ on the circle $\\abs{z} = r$ is small compared with $T(r, f)$ outside an exceptional set.\nIf $f$ is not rational, then $\\log r = o(T(r, f))$, so $S(r, f) = o(T(r, f))$ outside $E$.\nIf $f$ has finite order, the exceptional set $E$ can be omitted.\nFor transcendental $f$, dividing by $T(r, f)$ and using the [[first main theorem|QJWCFAK4AK]] gives the defect relation $\\sum_a \\delta(a) \\le 2$, where the sum runs over all points of the Riemann sphere.\nFor example, $e^z$ has $\\delta(0) = \\delta(\\infty) = 1$, so the relation is sharp.\n\n## Explanations\n\n### L1 human\n\nThe first main theorem says that no value is taken too often.\nThe second main theorem says the opposite for almost all values: a meromorphic function takes nearly every value about as often as its growth allows.\nAt most two values can be strongly avoided.\nThis is a quantitative form of Picard's theorem.\n\n### L2 human\n\n[[Picard's little theorem|R2PHM118NG]] follows at once: if a transcendental meromorphic $f$ omitted three values, the right side would be $S(r, f) = o(T(r, f))$ while the left side is at least $T(r, f)$.\nThe theorem also implies that the set of values $a$ with $\\delta(a) \u003e 0$ is countable.\nCounting without multiplicity yields results on ramification, such as bounds on the number of totally ramified values.\n[[Ahlfors theory|M8611CV76A]] gives a geometric proof of a similar inequality, based on covering surfaces.\n",
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