| import Mathlib |
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| namespace OAI |
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| /-! Primary Eisenstein elements, residue characters, Gauss sums and first-moment target definitions. -/ |
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| section |
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| noncomputable section |
| open scoped BigOperators |
| open Module |
| attribute [local instance] Classical.propDecidable |
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| namespace CubicFirstMoment |
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| def omega : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I / 3) |
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| def eisensteinRing : Subalgebra ℤ ℂ := Algebra.adjoin ℤ {omega} |
| abbrev Eisenstein := eisensteinRing |
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| def omegaE : Eisenstein := |
| ⟨omega, Algebra.subset_adjoin (Set.mem_singleton omega)⟩ |
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| def norm (a : Eisenstein) : ℝ := Complex.normSq (a : ℂ) |
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| def normNat (a : Eisenstein) : ℕ := ⌊norm a⌋₊ |
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| def primary (a : Eisenstein) : Prop := (3 : Eisenstein) ∣ a - 1 |
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| def primaryPrime (a : Eisenstein) : Prop := primary a ∧ Prime a |
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| def modulus (a : Eisenstein) : Ideal Eisenstein := Ideal.span {a} |
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| abbrev Residues (a : Eisenstein) := Eisenstein ⧸ modulus a |
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| def residueRepresentative (a : Eisenstein) (v : Residues a) : Eisenstein := |
| Classical.choose (Ideal.Quotient.mk_surjective v) |
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| def cubicSymbolAtPrime (p v : Eisenstein) : ℂ := |
| if IsUnit (Ideal.Quotient.mk (modulus p) v) then |
| if h : ∃ j : Fin 3, |
| Ideal.Quotient.mk (modulus p) (v ^ ((normNat p - 1) / 3)) = |
| Ideal.Quotient.mk (modulus p) (omegaE ^ (j : ℕ)) then |
| omega ^ ((Classical.choose h : Fin 3) : ℕ) |
| else 0 |
| else 0 |
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| def additivePhase (p v : Eisenstein) : ℂ := |
| let z : ℂ := (v : ℂ) / (p : ℂ) |
| Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (z + star z)) |
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| def gaussAtPrime (p : Eisenstein) : ℂ := |
| (Real.sqrt (norm p) : ℂ)⁻¹ * |
| ∑' v : Residues p, |
| cubicSymbolAtPrime p (residueRepresentative p v) * |
| additivePhase p (residueRepresentative p v) |
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| def cStar : ℝ := (2 * Real.pi) ^ (2 / 3 : ℝ) / (3 * Real.Gamma (2 / 3)) |
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| def primeCutoffSum (f : Eisenstein → ℂ) (X : ℝ) : ℂ := |
| ∑' p : Eisenstein, if primaryPrime p ∧ norm p ≤ X then f p else 0 |
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| def firstMomentScale (X : ℝ) : ℝ := X ^ (5 / 6 : ℝ) / Real.log X |
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| def FirstMomentStatement : Prop := |
| Asymptotics.IsLittleO Filter.atTop |
| (fun X : ℝ => primeCutoffSum gaussAtPrime X - |
| (((6 / 5 : ℝ) * cStar * firstMomentScale X : ℝ) : ℂ)) |
| firstMomentScale |
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| def theta (ℓ : ℤ) (a : Eisenstein) : ℂ := ((a : ℂ) / (‖(a : ℂ)‖ : ℂ)) ^ ℓ |
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| def AngularComparisonStatement : Prop := |
| ∀ ℓ : ℤ, Asymptotics.IsLittleO Filter.atTop |
| (primeCutoffSum (fun p => theta ℓ p * |
| (gaussAtPrime p - ((cStar * norm p ^ (-1 / 6 : ℝ) : ℝ) : ℂ)))) |
| firstMomentScale |
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| def AngularCancellationStatement : Prop := |
| ∀ ℓ : ℤ, ℓ ≠ 0 → Asymptotics.IsLittleO Filter.atTop |
| (primeCutoffSum (fun p => theta ℓ p * gaussAtPrime p)) firstMomentScale |
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|
| / |
| theorem patterson_firstMoment : FirstMomentStatement := by |
| sorry |
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| / |
| theorem patterson_angularComparison : AngularComparisonStatement := by |
| sorry |
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|
| / |
| theorem patterson_angularCancellation : AngularCancellationStatement := by |
| sorry |
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| end CubicFirstMoment |
| end |
| end |
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| end OAI |
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|
