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de504a9 | 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 | 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
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