| 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 |
|
|
