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8769036 | 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 | import OAI.NumberTheory.TwoPoint.Halasz.HalaszPrimeSupply
namespace OAI
/-! 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
open Finset Filter
open scoped Classical
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
end TwoPointCorrelations
end OAI
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