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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 | 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
end OAI
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