| import OAI.NumberTheory.TwoPoint.Halasz.HalaszPrimeSupply |
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| namespace OAI |
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| /-! The classical mean-value step using a finite supply of primes. |
| The absolute lower threshold for R comes only from Chebyshev's bounds. -/ |
| namespace TwoPointCorrelations |
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| open Finset Filter |
| open scoped Classical |
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| theorem halasz_classical_step : ∃ R₀ : ℕ, ∀ s k N R : ℕ, |
| 0<s → 2≤k → (4*(k:ℝ)^2)^2<(N:ℝ) → k<R → N<R^k → |
| R₀≤R → (2*(k^2*k)+1)^2≤R → |
| halaszVinogradovCount (k+s) k N ≤ |
| 2*(2*(k^2*k)+1)*((16*R)^(2*s)*((k^k*(16*R)^(k*(k-1)/2))* |
| (N^k*halaszVinogradovCount s k (N/R+1)))) := by |
| obtain ⟨R₀,hR₀⟩ := eventually_atTop.mp halasz_prime_supply_card |
| refine ⟨R₀,?_⟩ |
| intro s k N R hs hk hsize hkR hNR hR0 hmany |
| have hroot : ((2*(k^2*k)+1:ℕ):ℝ)≤Real.sqrt (R:ℝ) := by |
| apply Real.le_sqrt_of_sq_le |
| exact_mod_cast hmany |
| have hcard : 2*(k^2*k)+1≤(halaszPrimeSupply R).card := by |
| exact_mod_cast hroot.trans (hR₀ R hR0) |
| obtain ⟨P,hP,hPcard⟩ := exists_subset_card_eq hcard |
| have hb : ∀ p∈P, p.Prime ∧ R≤p ∧ p≤16*R := by |
| intro p hp |
| have hh := halasz_prime_supply_mem (hP hp) |
| exact ⟨hh.1,hh.2.1.le,hh.2.2⟩ |
| have h := halasz_vinogradov_prime_recurrence hs hk hsize hkR hNR P hb |
| (by rw [hPcard]; omega) |
| simpa only [hPcard] using h |
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| end TwoPointCorrelations |
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| end OAI |
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