The thread that runs through the whole rung
Look back at the four guides you have climbed. You learned to measure how fast an entire function grows with its order and type. You learned Jensen's formula, which says growth pays for zeros — every zero inside a disk charges a fee against the boundary average of log|f|. You factored finite-order functions over their zeros with Hadamard's theorem. And you met the little and great Picard theorems, which say a non-constant entire function misses at most one value, and near an essential singularity it takes every value but at most one infinitely often. Four guides, but really one question whispered four ways: how is the growth of a function related to how often it takes its values? Nevanlinna theory is the answer that holds all four in a single frame.
Here is the leap. Jensen counts only zeros — the places where f hits the single value 0. But 0 is nothing special; for any target w, the zeros of f(z) - w are the places where f hits w, and you could write Jensen's formula for that shifted function too. Rolf Nevanlinna's idea, around 1925, was to stop treating each value separately and instead build one measuring stick that compares the function's growth against its attainment of all values w simultaneously. That measuring stick is the characteristic function T(r, f), and the theory built on it, Nevanlinna theory, is the deepest thing in this rung.
Building the characteristic: how much, how high
The characteristic T(r, f) is built from two honest pieces, and you have already met both ideas in disguise. The first is the proximity function m(r, f): it measures how big f gets on the circle |z| = r, by averaging the positive part of log|f| around the circle. It only notices where |f| is large — where f is close to the value infinity. The second is the counting function N(r, f): it tallies the poles of f inside the disk |z| <= r, weighted by an integral of how the pole-count grows with radius — exactly the n(t)/t bookkeeping you saw squeeze out of Jensen's formula. Proximity says how close f comes to infinity on average; counting says how often it actually reaches infinity inside.
T(r, f) = m(r, f) + N(r, f)
m(r, f) = (1 / 2 pi) * integral from 0 to 2 pi of log+ |f(r e^(i theta))| d theta
( log+ x = max(log x, 0) : how CLOSE to infinity, on average )
N(r, f) = integral from 0 to r of ( n(t, f) - n(0, f) ) / t dt + n(0,f) log r
( n(t, f) = number of POLES in |z| <= t : how OFTEN infinity is reached )
For a value w : T(r, 1/(f - w)) counts how f attains the target w,
by reading the w-points of f as the POLES of 1/(f - w).Why glue m and N together into one number T? Because separately they are slippery, but their sum behaves like a clean growth gauge. T(r, f) is increasing in r and convex in log r, and for an entire function it sits right beside the maximum modulus: log M(r) and T(r, f) grow at the same rate, so the order you defined back in guide 1 can be read off from T just as well. The characteristic is the maximum modulus's more democratic cousin — instead of tracking only the biggest value of |f|, it folds in where the values go, and that is exactly the extra information value distribution needs.
The First Main Theorem: a Jensen for every value
Now the first big payoff, and it is Jensen's formula reborn. Apply the proximity-plus-counting split not to f but to 1/(f - w). The poles of 1/(f - w) are precisely the w-points of f, so its counting function N(r, 1/(f-w)) tallies how often f hits w inside the disk, and its proximity term m(r, 1/(f-w)) measures how close f comes to w on the circle. The First Main Theorem says: no matter which target w you choose, the total T(r, 1/(f-w)) is the same as T(r, f), up to a bounded wobble.
Written out, the statement is m(r, 1/(f-w)) + N(r, 1/(f-w)) = T(r, f) + O(1) for every finite value w. Read the two left-hand terms as the books: m is how close f comes to w on the circle, N is how often f actually equals w inside the disk, and their sum is pinned to the single growth budget T(r, f), give or take a bounded O(1). The budget is the same number for every w. That forces a trade-off: if f rarely equals w (so N is small), it must spend the rest of the budget hovering close to w (so m is large), and the reverse.
Sit with what this says, because it is genuinely beautiful. T(r, f) is a fixed growth budget. The First Main Theorem declares that every value w must, in total, be either attained or approached, and the two together always spend exactly that budget. A function with budget T cannot dodge a value for free: if it almost never equals w (so the counting term N is small), it is forced to spend the budget hovering close to w (so the proximity term m is large). Value-attainment is conserved. This is the precise sense in which Jensen's accounting of zeros was just the w = 0 page of a much thicker ledger — a Jensen's formula holding for every value at once.
The Second Main Theorem and the defect relation
The First Main Theorem is an equality, and equalities are even-handed: they let every value spend the budget however it likes — all on proximity, all on counting, or any mix. The deeper truth is that proximity is expensive and rare. The Second Main Theorem says that for almost every value, the function cannot afford to spend its budget on closeness; it is forced to spend it on actual attainment. Stack up several distinct targets w_1, ..., w_q and the theorem bounds the combined proximity by roughly 2 T(r, f) — a tiny allowance, far too small to cover q values if q is large. So most of the budget must go to N, to genuinely hitting the values.
The number that captures all this is the defect (or deficiency) of a value w. Define it as the fraction of the budget that w spends on proximity rather than attainment: delta(w) = liminf of m(r, 1/(f-w)) / T(r, f). A value with delta(w) = 0 is normal — f hits it about as often as the budget predicts. A value with delta(w) > 0 is deficient — f systematically hits it less often than its fair share, having to make up the difference by merely approaching it. A value f never hits at all (an exceptional value, like 0 for e^z) has the maximum defect delta(w) = 1.
Now the climax, the defect relation: when you add up the defects over all values w on the Riemann sphere (the finite values plus infinity), the total can be at most 2. In symbols, sum over all w of delta(w) <= 2. The whole sphere of possible target values has only two units of deficiency to share out. A function can pick a couple of favorites to slight, but it cannot slight three or more, because there simply is not enough defect to go around.
Picard, in one line, and the honest limits
Watch Picard fall out. An omitted value — one f never takes — has defect exactly 1, because all of its budget goes to proximity and none to counting. If a function omitted three distinct values, their defects would sum to at least 3, smashing the bound sum delta(w) <= 2. Impossible. So a non-constant meromorphic function can omit at most two values, and a non-constant entire function (whose omission of infinity is automatic, since it has no poles) can omit at most one finite value. That is the little Picard theorem — and here it is not a separate miracle but a one-line corollary of a counting inequality.
- Suppose, for contradiction, that a non-constant meromorphic f omits three distinct values w_1, w_2, w_3 on the sphere.
- An omitted value is never attained, so its counting term N(r, 1/(f - w_j)) is bounded; by the First Main Theorem its whole budget sits in proximity, giving defect delta(w_j) = 1.
- Sum the three: delta(w_1) + delta(w_2) + delta(w_3) = 3. But the defect relation forbids any total above 2.
- Contradiction. So no three values can be omitted: little Picard holds, and the great Picard theorem follows by applying the same machine near an essential singularity.
Be honest about what this glimpse does and does not give you. The defect relation packs Picard, the great Picard theorem, and far more into one inequality — but the proof of the Second Main Theorem is genuinely hard, leaning on a delicate 'lemma on the logarithmic derivative' that controls m(r, f'/f), and we have only sketched its consequences. The error terms hide real subtlety: the Second Main Theorem holds up to a small term that must be discarded on a thin set of radii r, which is why the defect relation is stated with a liminf. And the theory's reach has limits — it measures values in the plane and on the sphere, but it does not, for instance, settle questions about the Riemann zeta function's zeros; the Riemann hypothesis remains open and lies outside what value distribution alone can decide.
Step back and see the whole rung as one structure. Order and type gave you the scale of growth; Jensen tied that growth to the value 0; Hadamard rebuilt finite-order functions from their zeros; Picard exposed how few values a function can dodge; and Nevanlinna's characteristic gathers all of it into a single conserved budget that every value must spend. You came in able to say how fast a function grows. You leave able to say how that growth is distributed across the values it takes — which is the real subject hiding under the whole rung. That is the glimpse; the full theory of value distribution is one of the great landscapes of twentieth-century analysis, and you can now see its skyline.