| import OAI.NumberTheory.TwoPointCorrelations.MRTCountMean |
|
|
| namespace OAI |
|
|
| /-! Localize the Halasz distance hypothesis to the frequency window |
| actually used by a reciprocal-count cofactor. This allows a distant |
| frequency to be treated even when the minimizing twist is close to zero. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical ComplexConjugate |
|
|
| theorem halasz_local_count_prefix : ∃ C X₀ : ℝ, 0 < C ∧ |
| ∀ N : ℕ, X₀ ≤ N → |
| ∀ F : ℕ → ℂ, F 1 = 1 → Multiplicative F → OneBounded F → |
| ∀ Q P : Finset ℕ, (∀ p ∈ Q, p.Prime) → (∀ p ∈ P, p.Prime) → |
| ∀ t M : ℝ, 0 ≤ M → |
| (∀ v : ℝ, |v-t| ≤ Real.log (N:ℝ)^8 → |
| 2*M ≤ squaredDistance F (mrtArchimedeanTwist v) N) → |
| ‖∑ n ∈ Icc 1 N, (mrtMissingCoefficient F Q n * conj (mrtArchimedeanTwist t n)) / |
| ((finitePrimeDivisorCount P n : ℂ)+1)‖ ≤ |
| C*N*((M+1)*Real.exp (-M)+Real.log (Real.log N)/Real.log N) := by |
| obtain ⟨C, X₀, hC, hmean⟩ := halasz_general_mean_value |
| refine ⟨C, X₀, hC, ?_⟩ |
| intro N hN F hF1 hFm hFb Q P hQ hP t M hM hd |
| apply mrt_reciprocal_count_polynomial_bound (Icc 1 N) P |
| (fun n => mrtMissingCoefficient F Q n * conj (mrtArchimedeanTwist t n)) |
| intro u hu hu1 |
| let G := mrtCountMaskedCoefficient (mrtMissingCoefficient F Q) P u |
| have hG1 : G 1 = 1 := mrt_missing_count_one F hF1 Q P hQ hP u |
| have hGm : Multiplicative G := |
| (mrtMissingCoefficient_multiplicative F hFm Q hQ).mrtCountMaskedCoefficient P hP u |
| have hGb : OneBounded G := |
| (mrtMissingCoefficient_oneBounded F hFb Q).mrtCountMaskedCoefficient P hu hu1 |
| have hh := hmean N hN (halaszTwistedFunction G t) |
| (halasz_twisted_one G hG1 t) (halasz_twisted_multiplicative G hGm t) |
| (halasz_twisted_oneBounded G hGb t) M hM (by |
| intro v hv |
| rw [halasz_twisted_distance] |
| have hv' : |(t+v)-t| ≤ Real.log (N:ℝ)^8 := by |
| simpa only [add_sub_cancel_left] using abs_le.mpr hv |
| have hl := hd (t+v) hv' |
| have hm := mrt_missing_count_mask_distance F (mrtArchimedeanTwist (t+v)) Q P hu hu1 N |
| (fun p hp => hFb p ((mem_filter.mp hp).2.pos)) |
| (fun p _ => by rw [mrtArchimedeanTwist_norm]) |
| change squaredDistance F (mrtArchimedeanTwist (t+v)) N ≤ |
| 2*squaredDistance G (mrtArchimedeanTwist (t+v)) N at hm |
| linarith) |
| convert hh using 1 |
| congr 1 |
| apply sum_congr rfl |
| intro n _ |
| simp only [halaszTwistedFunction, G, mrtCountMaskedCoefficient] |
| ring |
|
|
| end TwoPointCorrelations |
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|
| end OAI |
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|
