| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterLogGrowth |
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| namespace OAI |
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| /-! Jensen's bound for the total multiplicity in the fixed normalized disk. |
| This counts actual zeros, with their analytic multiplicities, and will control |
| the entire logarithmic derivative once a zero-free strip supplies distance. -/ |
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| namespace TwoPointCorrelations |
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| open Complex Filter |
| open scoped BigOperators Classical Topology |
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| variable {q : ℕ} [NeZero q] |
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| theorem mrt_character_normalized_zero_count (χ : DirichletCharacter ℂ q) |
| (hχ : χ ≠ 1) (t : ℝ) : |
| (∑ ρ ∈ (mrtCharacterNormalizedZeros_finite χ hχ t).toFinset, |
| ((analyticOrderAt (mrtCharacterNormalizedLFunction χ t) ρ).toNat : ℝ)) ≤ |
| (1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) * |
| Real.log ((2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4)) := by |
| let f := mrtCharacterNormalizedLFunction χ t |
| have hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z := by |
| intro z _ |
| exact (mrtCharacterNormalizedLFunction_differentiable χ hχ t).analyticAt z |
| have hf0 : f 0 = 1 := mrtCharacterNormalizedLFunction_zero χ t |
| have hfactor : ∀ ρ : ℂ, ∃ g : ℂ → ℂ, AnalyticAt ℂ g ρ ∧ g ρ ≠ 0 ∧ |
| (ρ ∈ Erdos970.zerosetKfR (7 / 8) (by norm_num) f → |
| ∀ᶠ w in 𝓝 ρ, f w = (w - ρ) ^ (analyticOrderAt f ρ).toNat * g w) := by |
| intro ρ |
| by_cases hρ : ρ ∈ Erdos970.zerosetKfR (7 / 8) (by norm_num) f |
| · obtain ⟨g, hg, hgne, heq⟩ := Erdos970.lem_analytic_zero_factor |
| (15 / 16) (7 / 8) (by norm_num) (by norm_num) (by norm_num) f hf |
| (by rw [hf0]; exact one_ne_zero) ρ hρ |
| exact ⟨g, hg, hgne, fun _ => heq⟩ |
| · exact ⟨fun _ => 1, analyticAt_const, one_ne_zero, fun h => (hρ h).elim⟩ |
| let g : ℂ → ℂ → ℂ := fun ρ => Classical.choose (hfactor ρ) |
| have hg : ∀ ρ ∈ Erdos970.zerosetKfR (7 / 8) (by norm_num) f, |
| AnalyticAt ℂ (g ρ) ρ ∧ g ρ ρ ≠ 0 ∧ |
| ∀ᶠ w in 𝓝 ρ, f w = (w - ρ) ^ (analyticOrderAt f ρ).toNat * g ρ w := by |
| intro ρ hρ |
| exact ⟨(Classical.choose_spec (hfactor ρ)).1, |
| (Classical.choose_spec (hfactor ρ)).2.1, |
| (Classical.choose_spec (hfactor ρ)).2.2 hρ⟩ |
| have hK : 1 ≤ mrtCharacterInverseConstant := by |
| unfold mrtCharacterInverseConstant |
| exact le_add_of_nonneg_right (tsum_nonneg fun _ => norm_nonneg _) |
| have hB : 1 < (2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4) := by |
| have hq : (1 : ℝ) ≤ q := by exact_mod_cast NeZero.pos q |
| have hscale : 1 ≤ (2 * q : ℝ) * mrtCharacterInverseConstant := by nlinarith |
| have hm := le_mul_of_one_le_left (show 0 ≤ |t|+4 by positivity) hscale |
| linarith [abs_nonneg t] |
| exact Erdos970.lem_sum_m_rho_bound |
| ((2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4)) (15 / 16) (7 / 8) hB |
| (by norm_num) (by norm_num) (by norm_num) f hf |
| (by rw [hf0]; exact one_ne_zero) hf0 (mrtCharacterNormalizedZeros_finite χ hχ t) g |
| (fun z hz => mrtCharacterNormalizedLFunction_norm χ hχ t (hz.trans (by norm_num))) hg |
|
|
| /-- A distance lower bound controls every pole term at once. -/ |
| theorem mrt_character_normalized_logderiv_norm (χ : DirichletCharacter ℂ q) |
| (hχ : χ ≠ 1) (t : ℝ) {z : ℂ} (hz : ‖z‖ ≤ 3 / 4) |
| (hn : mrtCharacterNormalizedLFunction χ t z ≠ 0) {η : ℝ} (hη : 0 < η) |
| (hd : ∀ ρ ∈ mrtCharacterNormalizedZeros χ t, η ≤ ‖z - ρ‖) : |
| ‖deriv (mrtCharacterNormalizedLFunction χ t) z / mrtCharacterNormalizedLFunction χ t z‖ ≤ |
| (mrtCharacterLogDerivativeConstant + |
| (1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) / η) * |
| Real.log ((2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4)) := by |
| let S := (mrtCharacterNormalizedZeros_finite χ hχ t).toFinset |
| let m : ℂ → ℕ := fun ρ => (analyticOrderAt (mrtCharacterNormalizedLFunction χ t) ρ).toNat |
| have hsum : ‖∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)‖ ≤ (∑ ρ ∈ S, (m ρ : ℝ)) / η := by |
| calc |
| _ ≤ ∑ ρ ∈ S, ‖(m ρ : ℂ) / (z - ρ)‖ := norm_sum_le _ _ |
| _ ≤ ∑ ρ ∈ S, (m ρ : ℝ) / η := by |
| apply Finset.sum_le_sum |
| intro ρ hρ |
| rw [norm_div, Complex.norm_natCast] |
| exact div_le_div_of_nonneg_left (Nat.cast_nonneg _) hη |
| (hd ρ ((mrtCharacterNormalizedZeros_finite χ hχ t).mem_toFinset.mp hρ)) |
| _ = _ := by rw [Finset.sum_div] |
| have hcount := div_le_div_of_nonneg_right (mrt_character_normalized_zero_count χ hχ t) hη.le |
| have herr := mrtCharacterNormalized_logderiv χ hχ t hz hn |
| have htri := norm_add_le |
| (deriv (mrtCharacterNormalizedLFunction χ t) z / mrtCharacterNormalizedLFunction χ t z - |
| ∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)) |
| (∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)) |
| rw [sub_add_cancel] at htri |
| calc |
| _ ≤ mrtCharacterLogDerivativeConstant * |
| Real.log ((2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4)) + |
| ((1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) * |
| Real.log ((2 * q : ℝ) * mrtCharacterInverseConstant * (|t| + 4))) / η := by |
| exact htri.trans (add_le_add herr (hsum.trans hcount)) |
| _ = _ := by ring |
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| end TwoPointCorrelations |
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| end OAI |
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