Equations
- mellintransform2 = Lean.ParserDescr.node `mellintransform2 1024 (Lean.ParserDescr.symbol "𝓜")
Instances For
Equations
Instances For
@[reducible, inline]
Equations
- SmoothedChebyshevIntegrand SmoothingF ε X s = -deriv riemannZeta s / riemannZeta s * MellinTransform (fun (x : ℝ) => ↑(Smooth1 SmoothingF ε x)) s * ↑X ^ s
Instances For
Equations
- SmoothedChebyshev SmoothingF ε X = VerticalIntegral' (SmoothedChebyshevIntegrand SmoothingF ε X) (1 + (Real.log X)⁻¹)
Instances For
theorem
SmoothedChebyshevDirichlet_aux_integrable
{SmoothingF : ℝ → ℝ}
(diffSmoothingF : ContDiff ℝ 1 SmoothingF)
(SmoothingFpos : ∀ x > 0, 0 ≤ SmoothingF x)
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1)
{ε : ℝ}
(εpos : 0 < ε)
(ε_lt_one : ε < 1)
{σ : ℝ}
(σ_gt : 1 < σ)
(σ_le : σ ≤ 2)
:
MeasureTheory.Integrable
(fun (y : ℝ) => MellinTransform (fun (x : ℝ) => ↑(Smooth1 SmoothingF ε x)) (↑σ + ↑y * Complex.I)) MeasureTheory.volume
theorem
SmoothedChebyshevDirichlet_aux_tsum_integral
{SmoothingF : ℝ → ℝ}
(diffSmoothingF : ContDiff ℝ 1 SmoothingF)
(SmoothingFpos : ∀ x > 0, 0 ≤ SmoothingF x)
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1)
{X : ℝ}
(X_pos : 0 < X)
{ε : ℝ}
(εpos : 0 < ε)
(ε_lt_one : ε < 1)
{σ : ℝ}
(σ_gt : 1 < σ)
(σ_le : σ ≤ 2)
:
∫ (t : ℝ), ∑' (n : ℕ), ↑(ArithmeticFunction.vonMangoldt n) / ↑n ^ (↑σ + ↑t * Complex.I) * MellinTransform (fun (x : ℝ) => ↑(Smooth1 SmoothingF ε x)) (↑σ + ↑t * Complex.I) * ↑X ^ (↑σ + ↑t * Complex.I) = ∑' (n : ℕ), ∫ (t : ℝ), ↑(ArithmeticFunction.vonMangoldt n) / ↑n ^ (↑σ + ↑t * Complex.I) * MellinTransform (fun (x : ℝ) => ↑(Smooth1 SmoothingF ε x)) (↑σ + ↑t * Complex.I) * ↑X ^ (↑σ + ↑t * Complex.I)
theorem
SmoothedChebyshevDirichlet
{SmoothingF : ℝ → ℝ}
(diffSmoothingF : ContDiff ℝ 1 SmoothingF)
(SmoothingFpos : ∀ x > 0, 0 ≤ SmoothingF x)
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1)
{X : ℝ}
(X_gt : 3 < X)
{ε : ℝ}
(εpos : 0 < ε)
(ε_lt_one : ε < 1)
:
SmoothedChebyshev SmoothingF ε X = ↑(∑' (n : ℕ), ArithmeticFunction.vonMangoldt n * Smooth1 SmoothingF ε (↑n / X))
theorem
SmoothedChebyshevClose_aux
{Smooth1 : (ℝ → ℝ) → ℝ → ℝ → ℝ}
(SmoothingF : ℝ → ℝ)
(c₁ : ℝ)
(c₁_pos : 0 < c₁)
(c₁_lt : c₁ < 1)
(c₂ : ℝ)
(c₂_pos : 0 < c₂)
(c₂_lt : c₂ < 2)
(hc₂ : ∀ (ε x : ℝ), ε ∈ Set.Ioo 0 1 → 1 + c₂ * ε ≤ x → Smooth1 SmoothingF ε x = 0)
(C : ℝ)
(C_eq : C = 6 * (3 * c₁ + c₂))
(ε : ℝ)
(ε_pos : 0 < ε)
(ε_lt_one : ε < 1)
(X : ℝ)
(X_pos : 0 < X)
(X_gt_three : 3 < X)
(X_bound_1 : 1 ≤ X * ε * c₁)
(X_bound_2 : 1 ≤ X * ε * c₂)
(smooth1BddAbove : ∀ (n : ℕ), 0 < n → Smooth1 SmoothingF ε (↑n / X) ≤ 1)
(smooth1BddBelow : ∀ (n : ℕ), 0 < n → Smooth1 SmoothingF ε (↑n / X) ≥ 0)
(smoothIs1 : ∀ (n : ℕ), 0 < n → ↑n ≤ X * (1 - c₁ * ε) → Smooth1 SmoothingF ε (↑n / X) = 1)
(smoothIs0 : ∀ (n : ℕ), 1 + c₂ * ε ≤ ↑n / X → Smooth1 SmoothingF ε (↑n / X) = 0)
:
theorem
SmoothedChebyshevPull1_aux_integrable
{SmoothingF : ℝ → ℝ}
{ε : ℝ}
(ε_pos : 0 < ε)
(ε_lt_one : ε < 1)
{X : ℝ}
(X_gt : 3 < X)
{σ₀ : ℝ}
(σ₀_gt : 1 < σ₀)
(σ₀_le_2 : σ₀ ≤ 2)
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2)
(SmoothingFnonneg : ∀ x > 0, 0 ≤ SmoothingF x)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1)
(ContDiffSmoothingF : ContDiff ℝ 1 SmoothingF)
:
MeasureTheory.Integrable (fun (t : ℝ) => SmoothedChebyshevIntegrand SmoothingF ε X (↑σ₀ + ↑t * Complex.I))
MeasureTheory.volume
theorem
SmoothedChebyshevPull1
{SmoothingF : ℝ → ℝ}
{ε : ℝ}
(ε_pos : 0 < ε)
(ε_lt_one : ε < 1)
(X : ℝ)
(X_gt : 3 < X)
{T : ℝ}
(T_pos : 0 < T)
{σ₁ : ℝ}
(σ₁_pos : 0 < σ₁)
(σ₁_lt_one : σ₁ < 1)
(holoOn : HolomorphicOn (deriv riemannZeta / riemannZeta) (Set.Icc σ₁ 2 ×ℂ Set.Icc (-T) T \ {1}))
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2)
(SmoothingFnonneg : ∀ x > 0, 0 ≤ SmoothingF x)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1)
(ContDiffSmoothingF : ContDiff ℝ 1 SmoothingF)
:
theorem
SmoothedChebyshevPull2
{SmoothingF : ℝ → ℝ}
{ε : ℝ}
(ε_pos : 0 < ε)
(ε_lt_one : ε < 1)
(X : ℝ)
:
3 < X →
∀ {T : ℝ} (T_pos : 0 < T) {σ₁ σ₂ : ℝ} (σ₂_pos : 0 < σ₂) (σ₁_lt_one : σ₁ < 1) (σ₂_lt_σ₁ : σ₂ < σ₁)
(holoOn : HolomorphicOn (deriv riemannZeta / riemannZeta) (Set.Icc σ₁ 2 ×ℂ Set.Icc (-T) T \ {1}))
(holoOn2 : HolomorphicOn (SmoothedChebyshevIntegrand SmoothingF ε X) (Set.Icc σ₂ 2 ×ℂ Set.Icc (-3) 3 \ {1}))
(suppSmoothingF : Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2) (SmoothingFnonneg : ∀ x > 0, 0 ≤ SmoothingF x)
(mass_one : ∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1),
I₃₇ SmoothingF ε T X σ₁ = I₃ SmoothingF ε T X σ₁ - I₄ SmoothingF ε X σ₁ σ₂ + I₅ SmoothingF ε X σ₂ + I₆ SmoothingF ε X σ₁ σ₂ + I₇ SmoothingF ε T X σ₁
theorem
I1Bound :
∃ C > 0,
∀ {SmoothingF : ℝ → ℝ} {ε : ℝ},
0 < ε →
ε < 1 →
∀ (X : ℝ),
3 < X →
∀ {T : ℝ},
3 < T →
∀ {σ₁ : ℝ},
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF → ‖I₁ SmoothingF ε X T‖ ≤ C * X * Real.log X / (ε * T)
theorem
I9Bound :
∃ C > 0,
∀ {SmoothingF : ℝ → ℝ} {ε : ℝ},
0 < ε →
ε < 1 →
∀ (X : ℝ),
3 < X →
∀ {T : ℝ},
3 < T →
∀ {σ₁ : ℝ},
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF → ‖I₉ SmoothingF ε X T‖ ≤ C * X * Real.log X / (ε * T)
theorem
I2Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₂ SmoothingF ε X T σ₁‖ ≤ C * X / (ε * T)
theorem
I8Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₈ SmoothingF ε X T σ₁‖ ≤ C * X / (ε * T)
theorem
I3Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₃ SmoothingF ε X T σ₁‖ ≤ C * X * X ^ (-A / Real.log T ^ 9) / ε
theorem
I7Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₇ SmoothingF ε X T σ₁‖ ≤ C * X * X ^ (-A / Real.log T ^ 9) / ε
theorem
I4Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)) (σ₂ : ℝ) (_ : σ₂ ∈ Set.Ioo 0 1),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₄ SmoothingF ε X σ₁ σ₂‖ ≤ C * X * X ^ (-A / Real.log T ^ 9) / ε
theorem
I6Bound :
∃ (C : ℝ) (_ : 0 < C) (A : ℝ) (_ : A ∈ Set.Ioo 0 (1 / 2)) (σ₂ : ℝ) (_ : σ₂ ∈ Set.Ioo 0 1),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
∀ {T : ℝ},
3 < T →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF →
let σ₁ := 1 - A / Real.log X ^ 9;
‖I₆ SmoothingF ε X σ₁ σ₂‖ ≤ C * X * X ^ (-A / Real.log T ^ 9) / ε
theorem
I5Bound :
∃ (C : ℝ) (_ : 0 < C) (σ₂ : ℝ) (_ : σ₂ ∈ Set.Ioo 0 1),
∀ {SmoothingF : ℝ → ℝ} (X : ℝ),
3 < X →
∀ {ε : ℝ},
0 < ε →
ε < 1 →
Function.support SmoothingF ⊆ Set.Icc (1 / 2) 2 →
(∀ x > 0, 0 ≤ SmoothingF x) →
∫ (x : ℝ) in Set.Ioi 0, SmoothingF x / x = 1 →
ContDiff ℝ 1 SmoothingF → ‖I₅ SmoothingF ε X σ₂‖ ≤ C * X ^ σ₂ / ε