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  "body": "\n## Statement\n\nLet $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\\C$ that is not constant, and let $a \\in \\C$.\nFor $r \u003e 0$, let $m(r, a)$ and $N(r, a)$ be the proximity function and the counting function of $f$ at the value $a$, as in [[Nevanlinna theory|1EHH63VNAD]].\nLet $T(r, f)$ be the [[Nevanlinna characteristic|YT6Z0GWS0W]] of $f$.\nThen\n$$m(r, a) + N(r, a) = T(r, f) + O(1) \\qquad (r \\to \\infty),$$\nwhere the bounded term depends on $f$ and $a$ but not on $r$.\nMore precisely, if $f(0) \\ne a$ and $0$ is not a pole of $f$, then\n$$\\abs{\\, m(r, a) + N(r, a) - T(r, f) + \\log \\abs{f(0) - a} \\,} \\le \\log^+ \\abs{a} + \\log 2 \\quad \\text{for all } r \u003e 0.$$\n\n## Remarks\n\nApply [[Jensen's formula|497K99BQ4Y]] to $g = f - a$.\nIn Nevanlinna's notation it reads $T(r, g) - T(r, 1/g) = \\log \\abs{g(0)}$, and $T(r, 1/g) = m(r, a) + N(r, a)$.\nThe inequalities $\\log^+ \\abs{u - a} \\le \\log^+ \\abs{u} + \\log^+ \\abs{a} + \\log 2$ and its symmetric form show that $T(r, f - a)$ and $T(r, f)$ differ by at most $\\log^+ \\abs{a} + \\log 2$.\nFor example, for $f(z) = e^z$ and $a = 0$ there are no solutions of $f(z) = 0$, so $N(r, 0) = 0$, and $m(r, 0) = r/\\pi = T(r, f)$.\n\n## Explanations\n\n### L1 human\n\nFor each value $a$, the quantity $m(r, a) + N(r, a)$ measures how much $f$ is attracted to $a$ on the disc of radius $r$.\nThe counting part $N$ comes from the points where $f$ takes the value $a$.\nThe proximity part $m$ comes from the places on the circle where $f$ is very close to $a$.\nThe theorem says that the total is essentially the same for every value $a$, namely $T(r, f)$.\n\n### L2 human\n\nThe first main theorem is a reformulation of Jensen's formula, and it is easy compared with the [[second main theorem|GWYE6745JK]].\nIt gives the upper bound $N(r, a) \\le T(r, f) + O(1)$, which says that no value is taken too often.\nIt also justifies the definition of the defect $\\delta(a) = 1 - \\limsup_{r \\to \\infty} N(r, a)/T(r, f)$, which lies between $0$ and $1$.\nThe second main theorem gives the much deeper lower bound for most values.\n",
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