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465b10d | 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 | import Mathlib
namespace OAI
/-! Primary Eisenstein elements, residue characters, Gauss sums and first-moment target definitions. -/
section
noncomputable section
open scoped BigOperators
open Module
attribute [local instance] Classical.propDecidable
namespace CubicFirstMoment
def omega : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I / 3)
def eisensteinRing : Subalgebra ℤ ℂ := Algebra.adjoin ℤ {omega}
abbrev Eisenstein := eisensteinRing
def omegaE : Eisenstein :=
⟨omega, Algebra.subset_adjoin (Set.mem_singleton omega)⟩
def norm (a : Eisenstein) : ℝ := Complex.normSq (a : ℂ)
def normNat (a : Eisenstein) : ℕ := ⌊norm a⌋₊
def primary (a : Eisenstein) : Prop := (3 : Eisenstein) ∣ a - 1
def primaryPrime (a : Eisenstein) : Prop := primary a ∧ Prime a
def modulus (a : Eisenstein) : Ideal Eisenstein := Ideal.span {a}
abbrev Residues (a : Eisenstein) := Eisenstein ⧸ modulus a
def residueRepresentative (a : Eisenstein) (v : Residues a) : Eisenstein :=
Classical.choose (Ideal.Quotient.mk_surjective v)
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
def additivePhase (p v : Eisenstein) : ℂ :=
let z : ℂ := (v : ℂ) / (p : ℂ)
Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (z + star z))
def gaussAtPrime (p : Eisenstein) : ℂ :=
(Real.sqrt (norm p) : ℂ)⁻¹ *
∑' v : Residues p,
cubicSymbolAtPrime p (residueRepresentative p v) *
additivePhase p (residueRepresentative p v)
def cStar : ℝ := (2 * Real.pi) ^ (2 / 3 : ℝ) / (3 * Real.Gamma (2 / 3))
def primeCutoffSum (f : Eisenstein → ℂ) (X : ℝ) : ℂ :=
∑' p : Eisenstein, if primaryPrime p ∧ norm p ≤ X then f p else 0
def firstMomentScale (X : ℝ) : ℝ := X ^ (5 / 6 : ℝ) / Real.log X
def FirstMomentStatement : Prop :=
Asymptotics.IsLittleO Filter.atTop
(fun X : ℝ => primeCutoffSum gaussAtPrime X -
(((6 / 5 : ℝ) * cStar * firstMomentScale X : ℝ) : ℂ))
firstMomentScale
def theta (ℓ : ℤ) (a : Eisenstein) : ℂ := ((a : ℂ) / (‖(a : ℂ)‖ : ℂ)) ^ ℓ
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
def AngularCancellationStatement : Prop :=
∀ ℓ : ℤ, ℓ ≠ 0 → Asymptotics.IsLittleO Filter.atTop
(primeCutoffSum (fun p => theta ℓ p * gaussAtPrime p)) firstMomentScale
/-- The sharp first moment over all primary Eisenstein primes. -/
theorem patterson_firstMoment : FirstMomentStatement := by
sorry
/-- Model comparison in every fixed integer angular mode. -/
theorem patterson_angularComparison : AngularComparisonStatement := by
sorry
/-- Cancellation at the first-moment scale for each nonzero angular mode. -/
theorem patterson_angularCancellation : AngularCancellationStatement := by
sorry
end CubicFirstMoment
end
end
end OAI
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