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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 | import OAI.NumberTheory.TwoPointCorrelations.HalaszPrimeTruncation
import OAI.NumberTheory.TwoPointCorrelations.HalaszInnerApproximation
namespace OAI
/-! Apply the prime convolution a second time, retaining the varying
denominator log(N/p). This is the adaptive triple sum in the short proof of
Halasz's theorem. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
noncomputable def halaszDoubleConvolution (f : ℕ → ℂ) (N : ℕ) (L : ℝ) : ℂ :=
∑ p ∈ mrtPrimeBand L ((N : ℝ) / 2),
(Real.log (p : ℝ) : ℂ) * f p *
(halaszPrimeConvolution f (N / p) / (Real.log ((N : ℝ) / p) : ℂ))
lemma halasz_divisor_cutoff {N p : ℕ} (hp : 0 < p) (hpN : 2 * p ≤ N) :
1 ≤ N / p ∧ ((N / p : ℕ) : ℝ) ≤ (N : ℝ) / p ∧
(N : ℝ) / p ≤ 2 * ((N / p : ℕ) : ℝ) := by
have hp0 : (0 : ℝ) < p := by exact_mod_cast hp
have hM : 1 ≤ N / p := (Nat.le_div_iff_mul_le hp).mpr (by omega)
refine ⟨hM, ?_, ?_⟩
· apply (le_div_iff₀ hp0).mpr
exact_mod_cast Nat.div_mul_le_self N p
· have hmod : ((N % p : ℕ) : ℝ) < p := by exact_mod_cast Nat.mod_lt N hp
have hid : ((N % p : ℕ) : ℝ) + (p : ℝ) * ((N / p : ℕ) : ℝ) = N := by
exact_mod_cast Nat.mod_add_div N p
have hM1 : (1 : ℝ) ≤ ((N / p : ℕ) : ℝ) := by exact_mod_cast hM
apply (div_le_iff₀ hp0).mpr
nlinarith
theorem halasz_double_convolution_error (f : ℕ → ℂ) (hf : OneBounded f)
(hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n)
(N : ℕ) {L : ℝ} (hL : 1 ≤ L) :
‖halaszPrimeBandConvolution f N L ((N : ℝ) / 2) -
halaszDoubleConvolution f N L‖ ≤
halaszInnerErrorConstant * (N : ℝ) *
∑ p ∈ mrtPrimeBand L ((N : ℝ) / 2),
Real.log (p : ℝ) / ((p : ℝ) * Real.log ((N : ℝ) / p)) := by
have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _
have hL0 : 0 ≤ L := by linarith
unfold halaszPrimeBandConvolution halaszDoubleConvolution halaszPrimeTerm
rw [← sum_sub_distrib, mul_sum]
apply (norm_sum_le _ _).trans
apply sum_le_sum
intro p hp
have hprime := mrtPrimeBand_prime hp
have hb := mrtPrimeBand_bounds hL0 (by positivity : (0 : ℝ) ≤ (N : ℝ) / 2) hp
have hpN : 2 * p ≤ N := by exact_mod_cast (show 2 * (p : ℝ) ≤ N by linarith)
obtain ⟨hM, hMx, hxM⟩ := halasz_divisor_cutoff hprime.pos hpN
have hp0 : (0 : ℝ) < p := by exact_mod_cast hprime.pos
have hx : 1 < (N : ℝ) / p := by
have hh : (2 : ℝ) ≤ (N : ℝ) / p := (le_div_iff₀ hp0).mpr (by linarith)
linarith
have hlog : 0 ≤ Real.log (p : ℝ) := Real.log_nonneg (by exact_mod_cast hprime.one_le)
rw [← mul_sub, norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg hlog]
calc
_ ≤ Real.log (p : ℝ) * 1 *
(halaszInnerErrorConstant * ((N : ℝ) / p) / Real.log ((N : ℝ) / p)) := by
gcongr
· exact hf p hprime.pos
· exact halasz_real_inner_approximation f hf hmul (N / p) hM hx hMx hxM
_ = _ := by ring
end TwoPointCorrelations
end OAI
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