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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterLogGrowth
namespace OAI
/-! 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. -/
namespace TwoPointCorrelations
open Complex Filter
open scoped BigOperators Classical Topology
variable {q : ℕ} [NeZero q]
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
end TwoPointCorrelations
end OAI
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