openAI-math-dataset / lean /OAI /NumberTheory /Ostmann /QuadraticSieveComplementCorrelationsBasic.lean
| import OAI.NumberTheory.Ostmann.QuadraticSieveDivisorCorrelation | |
| import OAI.NumberTheory.Ostmann.QuadraticSieveLeadingCorrelations | |
| import OAI.NumberTheory.Ostmann.QuadraticSieveLeadingIntegral | |
| namespace OAI | |
| namespace Ostmann.QuadraticSieve | |
| open ComplexConjugate MeasureTheory Set | |
| open scoped SchwartzMap FourierTransform ArithmeticFunction.Moebius | |
| noncomputable def complementaryPair (S : Finset ℕ) (a : ℕ → ℂ) (F : ℕ → ℂ) : ℂ := | |
| ∑ n ∈ S, ∑ t ∈ S, if Nat.Coprime n t then a n * conj (a t) * F (n*t) else 0 | |
| noncomputable def complementaryCorrelation (W : 𝓢(ℝ, ℂ)) (M : ℝ) (K Δ : ℕ) | |
| (S : Finset ℕ) (a : ℕ → ℂ) : ℂ := | |
| complementaryPair S a (complementarySchwartzSum W M K Δ) | |
| noncomputable def complementaryCorrelationMain (W : 𝓢(ℝ, ℂ)) (M : ℝ) (K Δ : ℕ) | |
| (S : Finset ℕ) (a : ℕ → ℂ) (X₁ X₂ L : ℕ → ℝ) : ℂ := | |
| complementaryPair S a (fun q => complementaryPoissonMain W M K Δ q X₁ X₂ L) | |
| theorem complementaryPair_add (S : Finset ℕ) (a F G : ℕ → ℂ) : | |
| complementaryPair S a (fun q => F q + G q) = | |
| complementaryPair S a F + complementaryPair S a G := by | |
| simp only [complementaryPair, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| apply Finset.sum_congr rfl | |
| intro t ht | |
| split_ifs <;> ring | |
| theorem complementaryPair_sub (S : Finset ℕ) (a F G : ℕ → ℂ) : | |
| complementaryPair S a (fun q => F q - G q) = | |
| complementaryPair S a F - complementaryPair S a G := by | |
| simp only [complementaryPair, ← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| apply Finset.sum_congr rfl | |
| intro t ht | |
| split_ifs <;> ring | |
| theorem complementaryPair_mul (S : Finset ℕ) (a F : ℕ → ℂ) (z : ℂ) : | |
| complementaryPair S a (fun q => z * F q) = z * complementaryPair S a F := by | |
| simp only [complementaryPair, Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| apply Finset.sum_congr rfl | |
| intro t ht | |
| split_ifs <;> ring | |
| theorem complementaryPair_sum {ι : Type*} (V : Finset ι) (S : Finset ℕ) | |
| (a : ℕ → ℂ) (F : ι → ℕ → ℂ) : | |
| complementaryPair S a (fun q => ∑ v ∈ V, F v q) = | |
| ∑ v ∈ V, complementaryPair S a (F v) := by | |
| classical | |
| have heq (n t : ℕ) : | |
| (if Nat.Coprime n t then a n * conj (a t) * (∑ v ∈ V, F v (n*t)) else 0) = | |
| ∑ v ∈ V, if Nat.Coprime n t then a n * conj (a t) * F v (n*t) else 0 := by | |
| by_cases hc : Nat.Coprime n t | |
| · simp only [ite_eq_left hc, Finset.mul_sum] | |
| · simp only [ite_eq_right hc, Finset.sum_const_zero] | |
| simp only [complementaryPair, heq] | |
| calc | |
| _ = ∑ n ∈ S, ∑ v ∈ V, ∑ t ∈ S, | |
| if Nat.Coprime n t then a n * conj (a t) * F v (n*t) else 0 := | |
| Finset.sum_congr rfl (fun n hn => Finset.sum_comm) | |
| _ = _ := Finset.sum_comm | |
| theorem complementaryPair_ite (S : Finset ℕ) (a F : ℕ → ℂ) (P : Prop) [Decidable P] : | |
| complementaryPair S a (fun q => if P then F q else 0) = | |
| if P then complementaryPair S a F else 0 := by | |
| by_cases h : P <;> simp [h, complementaryPair] | |
| theorem complementaryPair_leading (M : ℝ) (I : ℂ) (K Δ : ℕ) | |
| (S : Finset ℕ) (a : ℕ → ℂ) : | |
| (∑ v ∈ oddSquarefreeUpTo K, if Nat.Coprime v Δ then | |
| complementaryPair S a (fun q => I * (Real.sqrt (M/v) : ℂ) * | |
| ((Nat.totient (2*q*Δ) : ℂ) / (2*q*Δ : ℕ)) * (jacobiSym (v : ℤ) q : ℂ)) else 0) = | |
| complementLeadingCorrelation M I K Δ S a := by | |
| unfold complementLeadingCorrelation | |
| simp only [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro v hv | |
| by_cases hc : Nat.Coprime v Δ | |
| · simp only [ite_eq_left hc, complementaryPair] | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| apply Finset.sum_congr rfl | |
| intro t ht | |
| by_cases hnt : Nat.Coprime n t | |
| · rw [ite_eq_left hnt, ite_eq_left (show Nat.Coprime n t ∧ Nat.Coprime v Δ from ⟨hnt,hc⟩)] | |
| ring | |
| · rw [ite_eq_right hnt, ite_eq_right (show ¬ (Nat.Coprime n t ∧ Nat.Coprime v Δ) from fun h => hnt h.1)] | |
| ring | |
| · simp only [hc, and_false, ite_false, Finset.sum_const_zero, mul_zero] | |
| theorem complementaryPair_divisors (S : Finset ℕ) (a : ℕ → ℂ) | |
| (N Δ v : ℕ) (F : ℕ → ℂ) | |
| (hS : ∀ n ∈ S, 0 < n ∧ n ≤ N ∧ Nat.Coprime (2*Δ) n) : | |
| complementaryPair S a (fun q => (jacobiSym (v : ℤ) q : ℂ) * | |
| ∑ d ∈ (2*q*Δ).divisors, (μ d : ℂ) * F d) = | |
| ∑ r ∈ (2*Δ).divisors, ∑ d ∈ Finset.Icc 1 (N^2), | |
| (μ r : ℂ) * (μ d : ℂ) * F (r*d) * | |
| coprimeProductDivisorJacobiRow S S a (fun n => conj (a n)) d (v : ℤ) := by | |
| have heq (q : ℕ) : 2*q*Δ = (2*Δ)*q := by ring | |
| simp_rw [heq] | |
| simpa only [complementaryPair, mul_assoc] using | |
| coprime_fixed_divisor_correlation_exchange S S a (fun n => conj (a n)) N (2*Δ) | |
| (v : ℤ) F hS hS | |
| end Ostmann.QuadraticSieve | |
| end OAI | |