import Nevanlinna.holomorphic_JensenFormula

open Real

variable {f : ℂ → ℂ}
variable {h₁f : AnalyticOn ℂ f ⊤}
variable {h₂f : f 0 ≠ 0}

noncomputable def unintegratedCounting : ℝ → ℕ := by
  intro r
  let A := h₁f.mono (by tauto : Metric.closedBall (0 : ℂ) r ⊆ ⊤)
  exact ∑ᶠ z, (A.order z).toNat

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)