nevanlinna/Nevanlinna/firstMain.lean

import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.holomorphic_JensenFormula

open Real

variable (f : ℂ → ℂ)
variable {h₁f : MeromorphicOn f ⊤}
variable {h₂f : f 0 ≠ 0}



noncomputable def Nevanlinna.n : ℝ → ℤ := by
  intro r
  have : MeromorphicAt f z := by
    exact h₁f (sorryAx ℂ true) trivial
  exact ∑ᶠ z ∈ Metric.ball (0 : ℂ) r, (h₁f z trivial).order

noncomputable def integratedCounting : ℝ → ℝ := by
  intro r
  -- Issue here: for s on the boundary of the ball, zero comes out.
  exact ∑ᶠ s, (h₁f.order s).toNat * log (r * ‖s.1‖⁻¹)