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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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | import OAI.NumberTheory.TwoPointCorrelations.HalaszPrimeTheta
import OAI.NumberTheory.TwoPointCorrelations.MRTSmoothDistance
namespace OAI
/-! Prime discrepancy for the near-minimizing twist in MRT Appendix A.
The deleted-prime contribution is kept separate in Cauchy--Schwarz; this
retains the strict exponent needed by the renormalization error. -/
namespace TwoPointCorrelations
open Finset Filter
open scoped ComplexConjugate Classical
noncomputable def halaszPrimeDiscrepancy (F : ℕ → ℂ) (t : ℝ) (N : ℕ) : ℝ :=
∑ p ∈ primesUpTo N, ‖1 - F p * conj (mrtArchimedeanTwist t p)‖ / p
lemma halasz_one_sub_norm_sq {z : ℂ} (hz : ‖z‖ ≤ 1) :
‖1 - z‖ ^ 2 ≤ 2 * (1 - z.re) := by
have hs : ‖z‖ ^ 2 ≤ 1 := by nlinarith [norm_nonneg z]
simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re,
Complex.sub_im, Complex.one_re, Complex.one_im] at hs ⊢
nlinarith
lemma halasz_prime_unit_bound (F : ℕ → ℂ) (hF : OneBounded F)
(t : ℝ) {N p : ℕ} (hp : p ∈ primesUpTo N) :
‖F p * conj (mrtArchimedeanTwist t p)‖ ≤ 1 := by
rw [norm_mul, Complex.norm_conj, mrtArchimedeanTwist_norm, mul_one]
exact hF p (mem_filter.mp hp).2.pos
lemma halasz_masked_discrepancy_split (F : ℕ → ℂ) (Q : Finset ℕ)
(hQ : ∀ p ∈ Q, p.Prime) (t : ℝ) (N : ℕ) :
halaszPrimeDiscrepancy (mrtMissingCoefficient F Q) t N =
(∑ p ∈ (primesUpTo N).filter (fun p => p ∈ Q), (1 : ℝ) / p) +
∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q),
‖1 - F p * conj (mrtArchimedeanTwist t p)‖ / p := by
unfold halaszPrimeDiscrepancy
rw [← sum_filter_add_sum_filter_not (primesUpTo N) (fun p => p ∈ Q)]
congr 1 <;> apply sum_congr rfl
· intro p hp
obtain ⟨hpN, hpQ⟩ := mem_filter.mp hp
rw [mrtMissingCoefficient, mrtPrimeMask_prime Q hQ (mem_filter.mp hpN).2,
ite_eq_left hpQ]
simp
· intro p hp
obtain ⟨hpN, hpQ⟩ := mem_filter.mp hp
rw [mrtMissingCoefficient, mrtPrimeMask_prime Q hQ (mem_filter.mp hpN).2,
ite_eq_right hpQ]
simp
lemma halasz_undeleted_discrepancy_sq (F : ℕ → ℂ) (hF : OneBounded F)
(Q : Finset ℕ) (t : ℝ) (N : ℕ) :
(∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q),
‖1 - F p * conj (mrtArchimedeanTwist t p)‖ / p) ^ 2 ≤
2 * (∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q), (1 : ℝ) / p) *
squaredDistance F (mrtArchimedeanTwist t) N := by
let S := (primesUpTo N).filter (fun p => p ∉ Q)
have hdef (p : ℕ) (hp : p ∈ primesUpTo N) :
0 ≤ 1 - (F p * conj (mrtArchimedeanTwist t p)).re :=
sub_nonneg.mpr ((Complex.re_le_norm _).trans (halasz_prime_unit_bound F hF t hp))
have hh := sum_sq_le_sum_mul_sum_of_sq_le_mul S
(r := fun p => ‖1 - F p * conj (mrtArchimedeanTwist t p)‖ / (p : ℝ))
(f := fun p => (1 : ℝ) / p)
(g := fun p => 2 * (1 - (F p * conj (mrtArchimedeanTwist t p)).re) / p)
(fun _ _ => by positivity)
(fun p hp => div_nonneg (mul_nonneg (by norm_num) (hdef p (mem_filter.mp hp).1))
(Nat.cast_nonneg p)) (fun p hp => ?_)
· have hd : (∑ p ∈ S, (1 - (F p * conj (mrtArchimedeanTwist t p)).re) / p) ≤
squaredDistance F (mrtArchimedeanTwist t) N := by
apply sum_le_sum_of_subset_of_nonneg (filter_subset _ _)
intro p hp _
exact div_nonneg (hdef p hp) (Nat.cast_nonneg p)
have he : (∑ p ∈ S, 2 * (1 - (F p * conj (mrtArchimedeanTwist t p)).re) / p) =
2 * ∑ p ∈ S, (1 - (F p * conj (mrtArchimedeanTwist t p)).re) / p := by
rw [mul_sum]
apply sum_congr rfl
intro p _
ring
rw [he] at hh
exact hh.trans (by
have h := mul_le_mul_of_nonneg_left hd
(show 0 ≤ 2 * ∑ p ∈ S, (1 : ℝ) / p by positivity)
convert h using 1
ring)
· have hsq := halasz_one_sub_norm_sq (halasz_prime_unit_bound F hF t (mem_filter.mp hp).1)
rw [div_pow]
apply (div_le_div_of_nonneg_right hsq (sq_nonneg (p : ℝ))).trans_eq
ring
lemma halasz_near_discrepancy_numeric (L A B Z D : ℝ) (hL : 0 ≤ L)
(hAB : A ≤ B)
(hAhi : A ≤ 51 / 100 * L) (hBhi : B ≤ 101 / 100 * L)
(hD : D ≤ L / 8) (hZsq : Z ^ 2 ≤ 2 * (B - A) * D) :
A + Z ≤ 7 / 8 * L := by
have hm := mul_le_mul_of_nonneg_left hD (by linarith : 0 ≤ 2 * (B - A))
have hprod : 0 ≤ (A - 51 / 100 * L) * (A - 99 / 100 * L) :=
mul_nonneg_of_nonpos_of_nonpos (by linarith) (by linarith)
by_contra hh
have hz : 7 / 8 * L - A < Z := by linarith
have hz0 : 0 ≤ 7 / 8 * L - A := by linarith
have hs := mul_self_lt_mul_self hz0 hz
nlinarith
/-- The `7/8` prime-discrepancy estimate used for near-twist renormalization.
Only primes below the actual Appendix-A cutoff may be deleted. -/
theorem halasz_near_masked_discrepancy :
∀ᶠ N : ℕ in atTop, ∀ (F : ℕ → ℂ), OneBounded F →
∀ (Q : Finset ℕ), (∀ p ∈ Q, p.Prime) →
(∀ p ∈ Q, (p : ℝ) ≤ Real.exp (Real.sqrt (Real.log N))) →
∀ t : ℝ, squaredDistance F (mrtArchimedeanTwist t) N ≤
Real.log (Real.log N) / 8 →
halaszPrimeDiscrepancy (mrtMissingCoefficient F Q) t N ≤
7 / 8 * Real.log (Real.log N) := by
obtain ⟨C, hC⟩ := primeReciprocalInput
have hlog : Tendsto (fun N : ℕ => Real.log N) atTop atTop :=
Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop
have hll := (Real.tendsto_log_atTop.comp hlog).eventually
(eventually_ge_atTop (100 * |C|))
filter_upwards [hlog.eventually (eventually_ge_atTop (4 : ℝ)), hll] with N hl hllN
dsimp only [Function.comp_def] at hllN
intro F hF Q hQ hcut t hD
let L := Real.log (Real.log N)
let A := ∑ p ∈ (primesUpTo N).filter (fun p => p ∈ Q), (1 : ℝ) / p
let B := ∑ p ∈ primesUpTo N, (1 : ℝ) / p
let Z := ∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q),
‖1 - F p * conj (mrtArchimedeanTwist t p)‖ / p
have hL : 0 ≤ L := by dsimp [L]; linarith [abs_nonneg C]
have hCsmall : C ≤ L / 100 := by dsimp [L]; linarith [le_abs_self C]
have hN1 : 1 < (N : ℝ) := (Real.log_pos_iff (Nat.cast_nonneg N)).mp (by linarith)
have hN2 : (2 : ℝ) ≤ N := by
have : 2 ≤ N := by exact_mod_cast hN1
exact_mod_cast this
have hB : B ≤ 101 / 100 * L := by
have hh := (abs_le.mp (hC N hN2)).2
rw [mrt_sievePrimesUpTo_nat] at hh
dsimp [B, L]
linarith
have hsqrt : 2 ≤ Real.sqrt (Real.log N) := by
have hs := Real.sq_sqrt (by linarith : 0 ≤ Real.log N)
have hn := Real.sqrt_nonneg (Real.log N)
nlinarith
have hY2 : 2 ≤ Real.exp (Real.sqrt (Real.log N)) := by
linarith [Real.add_one_le_exp (Real.sqrt (Real.log N))]
have hA : A ≤ 51 / 100 * L := by
have hsub : (primesUpTo N).filter (fun p => p ∈ Q) ⊆
sievePrimesUpTo (Real.exp (Real.sqrt (Real.log N))) := by
intro p hp
obtain ⟨hpN, hpQ⟩ := mem_filter.mp hp
exact mem_filter.mpr ⟨mem_Iic.mpr ((Nat.le_floor_iff (by positivity)).mpr
(hcut p hpQ)), (mem_filter.mp hpN).2⟩
have ha := sum_le_sum_of_subset_of_nonneg hsub
(f := fun p : ℕ => (1 : ℝ) / p) (by intros; positivity)
have hb := (abs_le.mp (hC _ hY2)).2
rw [Real.log_exp, Real.log_sqrt (by linarith : 0 ≤ Real.log N)] at hb
dsimp [A, L] at *
linarith
have hmass : (∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q), (1 : ℝ) / p) = B - A := by
have hh := sum_filter_add_sum_filter_not (primesUpTo N) (fun p => p ∈ Q)
(fun p : ℕ => (1 : ℝ) / p)
dsimp [A, B]
linarith
have hAB : A ≤ B := by
have hh : 0 ≤ ∑ p ∈ (primesUpTo N).filter (fun p => p ∉ Q), (1 : ℝ) / p := by positivity
rw [hmass] at hh
linarith
have hZ := halasz_undeleted_discrepancy_sq F hF Q t N
rw [hmass] at hZ
have hh := halasz_near_discrepancy_numeric L A B Z
(squaredDistance F (mrtArchimedeanTwist t) N) hL hAB hA hB hD hZ
simpa only [halasz_masked_discrepancy_split F Q hQ t N, A, Z] using hh
end TwoPointCorrelations
end OAI
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