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Stefan Kebekus
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@ -30,27 +30,28 @@ noncomputable def MeromorphicOn.N_zero
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z, (max 0 ((hf.restrict r).divisor z)) * log (r * ‖z‖⁻¹)
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fun r ↦ ∑ᶠ z, (max 0 ((hf.restrict |r|).divisor z)) * log (r * ‖z‖⁻¹)
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noncomputable def MeromorphicOn.N_infty
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z, (max 0 (-((hf.restrict r).divisor z))) * log (r * ‖z‖⁻¹)
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fun r ↦ ∑ᶠ z, (max 0 (-((hf.restrict |r|).divisor z))) * log (r * ‖z‖⁻¹)
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theorem Nevanlinna_counting₀
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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hf.inv.N_infty = hf.N_zero := by
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funext r
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unfold MeromorphicOn.N_zero MeromorphicOn.N_infty
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let A := (hf.restrict r).divisor.finiteSupport (isCompact_closedBall 0 r)
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let A := (hf.restrict |r|).divisor.finiteSupport (isCompact_closedBall 0 |r|)
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repeat
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rw [finsum_eq_sum_of_support_subset (s := A.toFinset)]
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apply Finset.sum_congr rfl
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intro x hx
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congr
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rw [hf.restrict_inv r]
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let B := hf.restrict_inv |r|
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rw [MeromorphicOn.divisor_inv]
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simp
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--
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@ -67,9 +68,8 @@ theorem Nevanlinna_counting₀
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contrapose
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simp
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intro hx h₁x
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rw [hf.restrict_inv r] at h₁x
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have hh : MeromorphicOn f (Metric.closedBall 0 r) := hf.restrict r
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rw [hh.divisor_inv] at h₁x
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rw [MeromorphicOn.divisor_inv (hf.restrict |r|)] at h₁x
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simp at h₁x
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rw [hx] at h₁x
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tauto
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@ -78,13 +78,13 @@ theorem Nevanlinna_counting₀
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theorem Nevanlinna_counting
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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hf.N_zero - hf.N_infty = fun r ↦ ∑ᶠ z, ((hf.restrict r).divisor z) * log (r * ‖z‖⁻¹) := by
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hf.N_zero - hf.N_infty = fun r ↦ ∑ᶠ z, ((hf.restrict |r|).divisor z) * log (r * ‖z‖⁻¹) := by
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funext r
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simp only [Pi.sub_apply]
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unfold MeromorphicOn.N_zero MeromorphicOn.N_infty
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let A := (hf.restrict r).divisor.finiteSupport (isCompact_closedBall 0 r)
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let A := (hf.restrict |r|).divisor.finiteSupport (isCompact_closedBall 0 |r|)
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repeat
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rw [finsum_eq_sum_of_support_subset (s := A.toFinset)]
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rw [← Finset.sum_sub_distrib]
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@ -92,9 +92,9 @@ theorem Nevanlinna_counting
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congr
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funext x
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congr
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by_cases h : 0 ≤ (hf.restrict r).divisor x
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by_cases h : 0 ≤ (hf.restrict |r|).divisor x
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· simp [h]
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· have h' : 0 ≤ -((hf.restrict r).divisor x) := by
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· have h' : 0 ≤ -((hf.restrict |r|).divisor x) := by
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simp at h
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apply Int.le_neg_of_le_neg
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simp
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@ -154,7 +154,7 @@ theorem Nevanlinna_firstMain₁
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(h₁f : MeromorphicOn f ⊤)
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(h₂f : StronglyMeromorphicAt f 0)
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(h₃f : f 0 ≠ 0) :
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(fun r ↦ log ‖f 0‖) + h₁f.inv.T_infty = h₁f.T_infty := by
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(fun _ ↦ log ‖f 0‖) + h₁f.inv.T_infty = h₁f.T_infty := by
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rw [add_eq_of_eq_sub]
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unfold MeromorphicOn.T_infty
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@ -170,10 +170,55 @@ theorem Nevanlinna_firstMain₁
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simp
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rw [← Nevanlinna_proximity h₁f]
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have hr : 0 < r := by sorry
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by_cases h₁r : r = 0
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rw [h₁r]
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simp
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have : π⁻¹ * 2⁻¹ * (2 * π * log (Complex.abs (f 0))) = (π⁻¹ * (2⁻¹ * 2) * π) * log (Complex.abs (f 0)) := by
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ring
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rw [this]
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clear this
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simp [pi_ne_zero]
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by_cases hr : 0 < r
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let A := jensen hr f (h₁f.restrict r) h₂f h₃f
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simp at A
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rw [A]
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clear A
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simp
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have {A B : ℝ} : -A + B = B - A := by ring
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rw [this]
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have : |r| = r := by
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rw [← abs_of_pos hr]
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simp
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rw [this]
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-- case 0 < -r
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have h₂r : 0 < -r := by
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simp [h₁r, hr]
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by_contra hCon
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-- Assume ¬(r < 0), which means r >= 0
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push_neg at hCon
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-- Now h is r ≥ 0, so we split into cases
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rcases lt_or_eq_of_le hCon with h|h
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· tauto
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· tauto
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let A := jensen h₂r f (h₁f.restrict (-r)) h₂f h₃f
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simp at A
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rw [A]
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clear A
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simp
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have {A B : ℝ} : -A + B = B - A := by ring
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rw [this]
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congr 1
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congr 1
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let A := integrabl_congr_negRadius (f := (fun z ↦ log (Complex.abs (f z)))) (r := r)
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rw [A]
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have : |r| = -r := by
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rw [← abs_of_pos h₂r]
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simp
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rw [this]
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let A := jensen
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sorry
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theorem Nevanlinna_firstMain₂
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{f : ℂ → ℂ}
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@ -181,4 +226,5 @@ theorem Nevanlinna_firstMain₂
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{r : ℝ}
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(h₁f : MeromorphicOn f ⊤) :
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|(h₁f.T_infty r) - ((h₁f.sub (MeromorphicOn.const a)).T_infty r)| ≤ logpos ‖a‖ + log 2 := by
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sorry
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@ -96,7 +96,7 @@ theorem jensen₀
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by_contra hCon
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rw [← hCon] at hx
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simp at hx
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rw [← StronglyMeromorphicAt.order_eq_zero_iff] at h₂z
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rw [← (h₁f z h₁z).order_eq_zero_iff] at h₂z
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unfold MeromorphicOn.divisor at hx
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simp [h₁z] at hx
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tauto
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@ -112,7 +112,7 @@ theorem jensen₀
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by_contra hCon
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rw [← hCon] at hx
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simp at hx
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rw [← StronglyMeromorphicAt.order_eq_zero_iff] at h₂z
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rw [← (h₁f z h₁z).order_eq_zero_iff] at h₂z
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unfold MeromorphicOn.divisor at hx
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simp [h₁z] at hx
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tauto
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@ -198,62 +198,28 @@ theorem jensen₀
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-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
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intro i _
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apply IntervalIntegrable.const_mul
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--simp at this
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by_cases h₂i : ‖i‖ = R
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-- case pos
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sorry
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--exact int'₂ h₂i
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apply intervalIntegrable_logAbs_circleMap_sub_const
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linarith
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--
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-- case neg
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 R x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 R x - ↑i) :=
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have : (fun x => log (Complex.abs ( F (circleMap 0 R x)))) = log ∘ Complex.abs ∘ F ∘ circleMap 0 R :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero R x]
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simp
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linarith
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let A := h₃F ⟨(circleMap 0 R x), circleMap_mem_closedBall 0 (le_of_lt hR) x⟩
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exact A
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--
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (F (circleMap 0 R x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 R x) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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have : (circleMap 0 R x) ∈ (Metric.closedBall 0 R) := by
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simp
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rw [abs_le]
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simp [hR]
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exact le_of_lt hR
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exact h₃F ⟨(circleMap 0 R x), this⟩
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-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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apply ContinuousAt.comp
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apply DifferentiableAt.continuousAt (𝕜 := ℂ)
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apply AnalyticAt.differentiableAt
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apply h₂F (circleMap 0 R x)
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simp; rw [abs_le]; simp [hR]; exact le_of_lt hR
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-- ContinuousAt (fun x => circleMap 0 1 x) x
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apply Continuous.continuousAt
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apply continuous_circleMap
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let A := h₂F (circleMap 0 R x) (circleMap_mem_closedBall 0 (le_of_lt hR) x)
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apply A.continuousAt
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exact (continuous_circleMap 0 R).continuousAt
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-- IntervalIntegrable (fun x => ∑ᶠ (s : ℂ), ↑(↑⋯.divisor s) * log (Complex.abs (circleMap 0 1 x - s))) MeasureTheory.volume 0 (2 * π)
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--simp? at h₁G
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have h₁G' {z : ℂ} : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * log (Complex.abs (z - s))) ⊆ ↑h₃f.toFinset := by
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@ -273,9 +239,8 @@ theorem jensen₀
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by_cases h₂i : ‖i‖ = R
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-- case pos
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--exact int'₂ h₂i
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sorry
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apply intervalIntegrable_logAbs_circleMap_sub_const (Ne.symm (ne_of_lt hR))
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-- case neg
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--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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