PrimeNumberTheoremAnd

The genus-one logarithmic-derivative zero terms are summable away from the zero set whenever the genus-one divisor product is summable.

The genus-one logarithmic-derivative zero terms are summable away from the zero set whenever the genus-one divisor product is summable. -/ theorem summable_logDerivTerms_divisorZeroIndex₀_of_summable_inv_sq {f : ℂ → ℂ} {z : ℂ} (h_sum : Summable (fun p : divisorZeroIndex₀ f (Set.univ : Set ℂ) => ‖divisorZeroIndex₀_val p‖⁻¹ ^ (2 : ℕ))) (hz : ∀ p : divisorZeroIndex₀ f (Set.univ : Set ℂ), z ≠ divisorZeroIndex₀_val p) : Summable (fun p : divisorZeroIndex₀ f (Set.univ : Set ℂ) => 1 / (z - divisorZeroIndex₀_val p) + 1 / divisorZeroIndex₀_val p) := by let R : ℝ := max ‖z‖ 1 have hRpos : 0 < R := lt_of_lt_of_le (by norm_num : (0 : ℝ) < 1) (le_max_right _ ) have hzle : ‖z‖ ≤ R := le_max_left _ _ let u : divisorZeroIndex₀ f (Set.univ : Set ℂ) → ℝ := fun p => (2 * R) * (‖divisorZeroIndex₀_val p‖⁻¹ ^ (2 : ℕ)) have hu : Summable u := h_sum.mul_left (2 * R) refine hu.of_norm_bounded_eventually ? have h_big : ∀ᶠ p : divisorZeroIndex₀ f (Set.univ : Set ℂ) in Filter.cofinite, (2 * R : ℝ) < ‖divisorZeroIndex₀_val p‖ := by have hfin : ({p : divisorZeroIndex₀ f (Set.univ : Set ℂ) | ‖divisorZeroIndex₀_val p‖ ≤ 2 * R} : Set _).Finite := by have : Metric.closedBall (0 : ℂ) (2 * R) ⊆ (Set.univ : Set ℂ) := by simp exact divisorZeroIndex₀_norm_le_finite (f := f) (U := (Set.univ : Set ℂ)) (B := 2 * R) this have := hfin.eventually_cofinite_notMem filter_upwards [this] with p hp have : ¬ ‖divisorZeroIndex₀_val p‖ ≤ 2 * R := by simpa using hp exact lt_of_not_ge this filter_upwards [h_big] with p hp let a : ℂ := divisorZeroIndex₀_val p have ha0 : a ≠ 0 := divisorZeroIndex₀_val_ne_zero p have hza0 : z - a ≠ 0 := sub_ne_zero.mpr (hz p) have hterm : 1 / (z - a) + 1 / a = z / (a * (z - a)) := by field_simp [ha0, hza0] ring have htri : ‖a‖ ≤ ‖z‖ + ‖z - a‖ := by have hraw : ‖a‖ ≤ ‖z‖ + ‖a - z‖ := by have h := norm_add_le z (a - z) simpa [a, sub_eq_add_neg, add_assoc, add_left_comm, add_comm] using h simpa [norm_sub_rev] using hraw have hza_lower : ‖a‖ / 2 ≤ ‖z - a‖ := by nlinarith [htri, hzle, hp] have hnorm : ‖1 / (z - a) + 1 / a‖ ≤ (2 * R) * (‖a‖⁻¹ ^ (2 : ℕ)) := by rw [hterm, norm_div, norm_mul] have ha_norm_pos : 0 < ‖a‖ := norm_pos_iff.mpr ha0 have hza_norm_pos : 0 < ‖z - a‖ := norm_pos_iff.mpr hza0 rw [div_eq_mul_inv] calc ‖z‖ * (‖a‖ * ‖z - a‖)⁻¹ = ‖z‖ * ‖a‖⁻¹ * ‖z - a‖⁻¹ := by field_simp [ha_norm_pos.ne', hza_norm_pos.ne'] _ ≤ R * ‖a‖⁻¹ * ‖z - a‖⁻¹ := by gcongr _ ≤ R * ‖a‖⁻¹ * (2 * ‖a‖⁻¹) := by gcongr have hhalf_pos : 0 < ‖a‖ / 2 := by positivity have hinv : ‖z - a‖⁻¹ ≤ (‖a‖ / 2)⁻¹ := by simpa [one_div] using one_div_le_one_div_of_le hhalf_pos hza_lower have hhalf_inv : (‖a‖ / 2)⁻¹ = 2 * ‖a‖⁻¹ := by field_simp [ha_norm_pos.ne'] simpa [hhalf_inv] using hinv _ = (2 * R) * (‖a‖⁻¹ ^ (2 : ℕ)) := by ring simpa [u, a] using hnorm

theorem hasProdUniformlyOn_divisorCanonicalProduct_univ (m : ℕ) (f : ℂ → ℂ) {K : Set ℂ} (hK : IsCompact K) (h_sum : Summable (fun p : divisorZeroIndex₀ f (Set.univ : Set ℂ) => ‖divisorZeroIndex₀_val p‖⁻¹ ^ (m + 1))) : HasProdUniformlyOn (fun (p : divisorZeroIndex₀ f (Set.univ : Set ℂ)) (z : ℂ) => weierstrassFactor m (z / divisorZeroIndex₀_val p)) (divisorCanonicalProduct m f (Set.univ : Set ℂ)) K := by rcases (isBounded_iff_forall_norm_le.1 hK.isBounded) with ⟨R0, hR0⟩ set R : ℝ := max R0 1 have hRpos : 0 < R := lt_of_lt_of_le (by norm_num : (0 : ℝ) < 1) (le_max_right _ _) have hnormK : ∀ z ∈ K, ‖z‖ ≤ R := fun z hzK => le_trans (hR0 z hzK) (le_max_left _ _) let g : divisorZeroIndex₀ f (Set.univ : Set ℂ) → ℂ → ℂ := fun p z => weierstrassFactor m (z / divisorZeroIndex₀_val p) - 1 let u : divisorZeroIndex₀ f (Set.univ : Set ℂ) → ℝ := fun p => (4 * R ^ (m + 1)) * (‖divisorZeroIndex₀_val p‖⁻¹ ^ (m + 1)) have hu : Summable u := h_sum.mul_left (4 * R ^ (m + 1)) have h_big : ∀ᶠ p : divisorZeroIndex₀ f (Set.univ : Set ℂ) in Filter.cofinite, (2 * R : ℝ) < ‖divisorZeroIndex₀_val p‖ := by have hfin : ({p : divisorZeroIndex₀ f (Set.univ : Set ℂ) | ‖divisorZeroIndex₀_val p‖ ≤ 2 * R} : Set _).Finite := by have : Metric.closedBall (0 : ℂ) (2 * R) ⊆ (Set.univ : Set ℂ) := by simp exact divisorZeroIndex₀_norm_le_finite (f := f) (U := (Set.univ : Set ℂ)) (B := 2 * R) this have := hfin.eventually_cofinite_notMem filter_upwards [this] with p hp have : ¬ ‖divisorZeroIndex₀_val p‖ ≤ 2 * R := by simpa using hp exact lt_of_not_ge this have hBound : ∀ᶠ p in Filter.cofinite, ∀ z ∈ K, ‖g p z‖ ≤ u p := by filter_upwards [h_big] with p hp z hzK have hzle : ‖z‖ ≤ R := hnormK z hzK have hz_div : ‖z / divisorZeroIndex₀_val p‖ ≤ (1 / 2 : ℝ) := by exact norm_div_le_half_of_norm_le_of_two_mul_lt hRpos hzle hp have hE : ‖weierstrassFactor m (z / divisorZeroIndex₀_val p) - 1‖ ≤ 4 * ‖z / divisorZeroIndex₀_val p‖ ^ (m + 1) := weierstrassFactor_sub_one_pow_bound (m := m) (z := z / divisorZeroIndex₀_val p) hz_div have hz_pow : ‖z / divisorZeroIndex₀_val p‖ ^ (m + 1) ≤ (R ^ (m + 1)) * (‖divisorZeroIndex₀_val p‖⁻¹ ^ (m + 1)) := by have : ‖z / divisorZeroIndex₀_val p‖ = ‖z‖ * ‖divisorZeroIndex₀_val p‖⁻¹ := by simp [div_eq_mul_inv] rw [this] have : (‖z‖ * ‖divisorZeroIndex₀_val p‖⁻¹) ^ (m + 1) = ‖z‖ ^ (m + 1) * (‖divisorZeroIndex₀_val p‖⁻¹ ^ (m + 1)) := by simp [mul_pow] rw [this] have hzle_pow : ‖z‖ ^ (m + 1) ≤ R ^ (m + 1) := pow_le_pow_left₀ (norm_nonneg z) hzle (m + 1) gcongr dsimp [g, u] nlinarith [hE, hz_pow] have hcts : ∀ p, ContinuousOn (g p) K := by intro p have hcontE : Continuous (fun z : ℂ => weierstrassFactor m z) := (differentiable_weierstrassFactor m).continuous have hdiv : Continuous fun z : ℂ => z / divisorZeroIndex₀_val p := by simpa [div_eq_mul_inv] using! (continuous_id.mul continuous_const) have hcont : Continuous fun z : ℂ => weierstrassFactor m (z / divisorZeroIndex₀_val p) := hcontE.comp hdiv simpa only [g] using hcont.continuousOn.fun_sub continuous_const.continuousOn have hprod : HasProdUniformlyOn (fun p z ↦ 1 + g p z) (fun z ↦ ∏' p, (1 + g p z)) K := by simpa using Summable.hasProdUniformlyOn_one_add (f := g) (u := u) (K := K) hK hu hBound hcts simpa [g, divisorCanonicalProduct, sub_eq_add_neg, add_assoc, add_left_comm, add_comm] using! hprod

/-!

Entire-ness (holomorphy) of the divisor-indexed canonical product

The logarithmic derivative of a genus-one divisor canonical product is the expected sum of zero terms. The logarithmic-derivative term summability is derived from the genus-one divisor summability and the assumption that the evaluation point is not one of the indexed zeros.

A (nonzero) divisor index has nonzero multiplicity at its underlying point.