ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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A centre a-point requires regularised counting

Example

Let $a, c \in C$ , let $m \geq 1$ be an integer, and suppose $c \neq 0$ . Set $f (z) = a + c z^{m}$ . For every $r > 0$ , the central $a$ -point has multiplicity $m$ , so $n (0, a; f) = m$ and $N (r, a; f) = m \log r$ , although the unregularized integral $\int_{0}^{r} n (t, a; f) d t / t$ diverges. The centre Jensen mean is $M_{r} \log ∣ f - a ∣ = \log ∣ c ∣ + m \log r$ , and the exact First Main Theorem constant is $\frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ c ∣$ . The finite formulas use $c$ because $f (0) - a = 0$ .

Verification

Given: $a, c \in C$ , integer $m \geq 1$ , $c \neq 0$ , and $f (z) = a + c z^{m}$ .

[F1] For finite $w, a$ , $δ (w, a) = ∣ w - a ∣ / (\sqrt{1 + ∣ w ∣^{2}} \sqrt{1 + ∣ a ∣^{2}})$ ; $m (r, a; f)$ is the circular mean of $\log (1 / δ (f, a))$ , and $T (r, f) = m (r, \infty; f) + N (r, \infty; f)$ (Counting, chordal proximity and characteristic).

[F2] $N (r, b; f) = n (0, b; f) \log r + \int_{0}^{r} (n (t, b; f) - n (0, b; f)) d t / t$ (Counting, chordal proximity and characteristic).

[F3] For a finite target $b$ , $M_{r} \log ∣ f - b ∣ = \log ∣ c_{b} ∣ + N (r, b; f) - N (r, \infty; f)$ , where $c_{b}$ is the first nonzero Laurent coefficient of $f - b$ at $0$ (Meromorphic Jensen identity with a zero or pole at the centre).

[F4] For nonconstant meromorphic $f$ and finite $b$ , $m (r, b; f) + N (r, b; f) = T (r, f) + C (f, b)$ (Nevanlinna’s First Main Theorem with exact centre constant).

[F5] When $f (z) - b = c_{b} z^{k_{b}} + \dots$ at $0$ , the exact constant is $C (f, b) = \frac{1}{2} \log (1 + ∣ b ∣^{2}) - \log ∣ c_{b} ∣$ (Nevanlinna’s First Main Theorem with exact centre constant).

1.1givenalgebra

Since $f (z) - a = c z^{m}$ with $c \neq 0$ , its only $a$ -point is $0$ , of multiplicity $m$ , and it has no poles. Thus $n (t, a; f) = m$ for every $t \geq 0$ , while $n (t, \infty; f) = 0$ .

1.2F2algebra

Substituting these counts into [F2] gives $N (r, a; f) = m \log r + \int_{0}^{r} (m - m) d t / t = m \log r$ for every $r > 0$ . The central term is the whole regularized count.

1.3F2algebra

For $0 < ε < r$ , the unregularized integral from $ε$ to $r$ is $\int_{ε}^{r} n (t, a; f) d t / t = m \log (r / ε) \to + \infty$ as $ε ↓ 0$ . Thus $\int_{0}^{r} n (t, a; f) d t / t$ diverges for every $r > 0$ , even though the regularized $N (r, a; f)$ is finite.

2.1F3step 1.2algebra

On $∣ z ∣ = r$ , $\log ∣ f (z) - a ∣ = \log ∣ c ∣ + m \log r$ , so its circular mean is $\log ∣ c ∣ + m \log r$ . Since $N (r, \infty; f) = 0$ , [F3] gives $M_{r} \log ∣ f - a ∣ = \log ∣ c ∣ + N (r, a; f) = \log ∣ c ∣ + m \log r$ , agreeing with the direct boundary calculation.

3.1F1step 2.1algebra

By [F1], the finite-target chordal identity averages to $m (r, a; f) = T (r, f) + \frac{1}{2} \log (1 + ∣ a ∣^{2}) - M_{r} \log ∣ f - a ∣$ , because $f$ has no poles and hence $T (r, f) = m (r, \infty; f)$ . Using step 2.1 gives $m (r, a; f) + N (r, a; f) = T (r, f) + \frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ c ∣$ . This computes the finite-target constant directly.

4.1F3F4F5step 3.1algebra∎

Here $f (z) - a = c z^{m}$ , so $c_{a} = c$ and $k_{a} = m$ in [F3] and [F5]; the First Main Theorem constant is exactly $C (f, a) = \frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ c ∣$ , agreeing with step 3.1. Since $f (0) - a = 0$ , $\log ∣ f (0) - a ∣$ is not a finite logarithm and cannot replace the term $\log ∣ c ∣$ .

Depends on

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Sources

Alexandre Eremenko, Lectures on Nevanlinna Theory, §1, equation (2) and footnotes 1–2 (standard reference, not scraped)
Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §§2,4 (standard reference, not scraped)