Documentation

PrimeNumberTheoremAnd.MediumPNT

noncomputable def ChebyshevPsi (x : ) :
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    @[reducible, inline]
    noncomputable abbrev SmoothedChebyshevIntegrand (SmoothingF : ) (ε X : ) :
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      noncomputable def SmoothedChebyshev (SmoothingF : ) (ε X : ) :
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        theorem SmoothedChebyshevDirichlet_aux_integrable {SmoothingF : } (diffSmoothingF : ContDiff 1 SmoothingF) (SmoothingFpos : x > 0, 0 SmoothingF x) (suppSmoothingF : Function.support SmoothingF Set.Icc (1 / 2) 2) (mass_one : (x : ) in Set.Ioi 0, SmoothingF x / x = 1) {ε : } (εpos : 0 < ε) (ε_lt_one : ε < 1) {σ : } (σ_gt : 1 < σ) (σ_le : σ 2) :
        MeasureTheory.Integrable (fun (y : ) => MellinTransform (fun (x : ) => (Smooth1 SmoothingF ε x)) (σ + y * Complex.I)) MeasureTheory.volume
        theorem SmoothedChebyshevDirichlet_aux_tsum_integral {SmoothingF : } (diffSmoothingF : ContDiff 1 SmoothingF) (SmoothingFpos : x > 0, 0 SmoothingF x) (suppSmoothingF : Function.support SmoothingF Set.Icc (1 / 2) 2) (mass_one : (x : ) in Set.Ioi 0, SmoothingF x / x = 1) {X : } (X_pos : 0 < X) {ε : } (εpos : 0 < ε) (ε_lt_one : ε < 1) {σ : } (σ_gt : 1 < σ) (σ_le : σ 2) :
        (t : ), ∑' (n : ), (ArithmeticFunction.vonMangoldt n) / n ^ (σ + t * Complex.I) * MellinTransform (fun (x : ) => (Smooth1 SmoothingF ε x)) (σ + t * Complex.I) * X ^ (σ + t * Complex.I) = ∑' (n : ), (t : ), (ArithmeticFunction.vonMangoldt n) / n ^ (σ + t * Complex.I) * MellinTransform (fun (x : ) => (Smooth1 SmoothingF ε x)) (σ + t * Complex.I) * X ^ (σ + t * Complex.I)
        theorem SmoothedChebyshevDirichlet {SmoothingF : } (diffSmoothingF : ContDiff 1 SmoothingF) (SmoothingFpos : x > 0, 0 SmoothingF x) (suppSmoothingF : Function.support SmoothingF Set.Icc (1 / 2) 2) (mass_one : (x : ) in Set.Ioi 0, SmoothingF x / x = 1) {X : } (X_gt : 3 < X) {ε : } (εpos : 0 < ε) (ε_lt_one : ε < 1) :
        SmoothedChebyshev SmoothingF ε X = (∑' (n : ), ArithmeticFunction.vonMangoldt n * Smooth1 SmoothingF ε (n / X))
        theorem SmoothedChebyshevClose_aux {Smooth1 : ()} (SmoothingF : ) (c₁ : ) (c₁_pos : 0 < c₁) (c₁_lt : c₁ < 1) (c₂ : ) (c₂_pos : 0 < c₂) (c₂_lt : c₂ < 2) (hc₂ : ∀ (ε x : ), ε Set.Ioo 0 11 + c₂ * ε xSmooth1 SmoothingF ε x = 0) (C : ) (C_eq : C = 6 * (3 * c₁ + c₂)) (ε : ) (ε_pos : 0 < ε) (ε_lt_one : ε < 1) (X : ) (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 : ), 0 < nSmooth1 SmoothingF ε (n / X) 1) (smooth1BddBelow : ∀ (n : ), 0 < nSmooth1 SmoothingF ε (n / X) 0) (smoothIs1 : ∀ (n : ), 0 < nn X * (1 - c₁ * ε) → Smooth1 SmoothingF ε (n / X) = 1) (smoothIs0 : ∀ (n : ), 1 + c₂ * ε n / XSmooth1 SmoothingF ε (n / X) = 0) :
        (∑' (n : ), ArithmeticFunction.vonMangoldt n * Smooth1 SmoothingF ε (n / X)) - ((Finset.range X + 1⌋₊).sum ArithmeticFunction.vonMangoldt) C * ε * X * Real.log X
        theorem SmoothedChebyshevClose {SmoothingF : } (diffSmoothingF : ContDiff 1 SmoothingF) (suppSmoothingF : Function.support SmoothingF Set.Icc (1 / 2) 2) (SmoothingFnonneg : x > 0, 0 SmoothingF x) (mass_one : (x : ) in Set.Ioi 0, SmoothingF x / x = 1) :
        ∃ (C : ), ∀ (X : ), 3 < X∀ (ε : ), 0 < εε < 12 < X * εSmoothedChebyshev SmoothingF ε X - (ChebyshevPsi X) C * ε * X * Real.log X
        noncomputable def I₁ (SmoothingF : ) (ε X T : ) :
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          noncomputable def I₂ (SmoothingF : ) (ε T X σ₁ : ) :
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            noncomputable def I₃₇ (SmoothingF : ) (ε T X σ₁ : ) :
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              noncomputable def I₈ (SmoothingF : ) (ε T X σ₁ : ) :
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                noncomputable def I₉ (SmoothingF : ) (ε X T : ) :
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                  noncomputable def I₃ (SmoothingF : ) (ε T X σ₁ : ) :
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                    noncomputable def I₇ (SmoothingF : ) (ε T X σ₁ : ) :
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                      noncomputable def I₄ (SmoothingF : ) (ε X σ₁ σ₂ : ) :
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                        noncomputable def I₆ (SmoothingF : ) (ε X σ₁ σ₂ : ) :
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                          noncomputable def I₅ (SmoothingF : ) (ε X σ₂ : ) :
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                            theorem realDiff_of_complexDIff {f : } (s : ) (hf : DifferentiableAt f s) :
                            ContinuousAt (fun (x : ) => f (s.re + x * Complex.I)) s.im
                            theorem riemannZeta_bdd_on_vertical_lines {σ₀ : } (σ₀_gt : 1 < σ₀) (t : ) :
                            c > 0, riemannZeta (σ₀ + t * Complex.I) c
                            theorem summable_real_iff_summable_coe_complex (f : ) :
                            Summable f Summable fun (n : ) => (f n)
                            theorem cast_pow_eq (n : ) (σ₀ : ) :
                            ↑(n ^ σ₀) = n ^ σ₀
                            theorem dlog_riemannZeta_bdd_on_vertical_lines {σ₀ : } (σ₀_gt : 1 < σ₀) :
                            c > 0, ∀ (t : ), deriv riemannZeta (σ₀ + t * Complex.I) / riemannZeta (σ₀ + t * Complex.I) c
                            theorem analyticAt_riemannZeta {s : } (s_ne_one : s 1) :
                            theorem dlog_riemannZeta_bdd_on_vertical_lines' {σ₀ : } (σ₀_gt : 1 < σ₀) :
                            C > 0, ∀ (t : ), deriv riemannZeta (σ₀ + t * Complex.I) / riemannZeta (σ₀ + t * Complex.I) C
                            theorem SmoothedChebyshevPull1_aux_integrable {SmoothingF : } {ε : } (ε_pos : 0 < ε) (ε_lt_one : ε < 1) {X : } (X_gt : 3 < X) {σ₀ : } (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 : ) in Set.Ioi 0, SmoothingF x / x = 1) (ContDiffSmoothingF : ContDiff 1 SmoothingF) :
                            theorem BddAboveOnRect {g : } {z w : } (holoOn : HolomorphicOn g (z.Rectangle w)) :
                            theorem SmoothedChebyshevPull1 {SmoothingF : } {ε : } (ε_pos : 0 < ε) (ε_lt_one : ε < 1) (X : ) (X_gt : 3 < X) {T : } (T_pos : 0 < T) {σ₁ : } (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 : ) in Set.Ioi 0, SmoothingF x / x = 1) (ContDiffSmoothingF : ContDiff 1 SmoothingF) :
                            SmoothedChebyshev SmoothingF ε X = I₁ SmoothingF ε X T - I₂ SmoothingF ε T X σ₁ + I₃₇ SmoothingF ε T X σ₁ + I₈ SmoothingF ε T X σ₁ + I₉ SmoothingF ε X T + MellinTransform (fun (x : ) => (Smooth1 SmoothingF ε x)) 1 * X
                            theorem SmoothedChebyshevPull2 {SmoothingF : } {ε : } (ε_pos : 0 < ε) (ε_lt_one : ε < 1) (X : ) :
                            3 < X∀ {T : } (T_pos : 0 < T) {σ₁ σ₂ : } (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 : ) 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 ZetaBoxEval {SmoothingF : } (suppSmoothingF : Function.support SmoothingF Set.Icc (1 / 2) 2) (mass_one : (x : ) in Set.Ioi 0, SmoothingF x / x = 1) (ContDiffSmoothingF : ContDiff 1 SmoothingF) :
                            ∃ (C : ), ∀ᶠ (ε : ) in nhdsWithin 0 (Set.Ioi 0), ∀ (X : ), 0 XMellinTransform (fun (x : ) => (Smooth1 SmoothingF ε x)) 1 * X - X C * ε * X
                            theorem IBound_aux1 {k : } (k_pos : 0 < k) :
                            C > 0, ∀ {T : }, 3 < TReal.log T ^ k C * T
                            theorem I1Bound :
                            C > 0, ∀ {SmoothingF : } {ε : }, 0 < εε < 1∀ (X : ), 3 < X∀ {T : }, 3 < T∀ {σ₁ : }, Function.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFI₁ SmoothingF ε X T C * X * Real.log X / (ε * T)
                            theorem I9Bound :
                            C > 0, ∀ {SmoothingF : } {ε : }, 0 < εε < 1∀ (X : ), 3 < X∀ {T : }, 3 < T∀ {σ₁ : }, Function.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFI₉ SmoothingF ε X T C * X * Real.log X / (ε * T)
                            theorem I2Bound :
                            ∃ (C : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet σ₁ := 1 - A / Real.log X ^ 9; I₂ SmoothingF ε X T σ₁ C * X / (ε * T)
                            theorem I8Bound :
                            ∃ (C : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet σ₁ := 1 - A / Real.log X ^ 9; I₈ SmoothingF ε X T σ₁ C * X / (ε * T)
                            theorem I3Bound :
                            ∃ (C : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet 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 : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet 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 : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)) (σ₂ : ) (_ : σ₂ Set.Ioo 0 1), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet 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 : ) (_ : 0 < C) (A : ) (_ : A Set.Ioo 0 (1 / 2)) (σ₂ : ) (_ : σ₂ Set.Ioo 0 1), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1∀ {T : }, 3 < TFunction.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFlet 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 : ) (_ : 0 < C) (σ₂ : ) (_ : σ₂ Set.Ioo 0 1), ∀ {SmoothingF : } (X : ), 3 < X∀ {ε : }, 0 < εε < 1Function.support SmoothingF Set.Icc (1 / 2) 2(∀ x > 0, 0 SmoothingF x) (x : ) in Set.Ioi 0, SmoothingF x / x = 1ContDiff 1 SmoothingFI₅ SmoothingF ε X σ₂ C * X ^ σ₂ / ε
                            theorem MediumPNT :
                            c > 0, (ChebyshevPsi - id) =O[Filter.atTop] fun (x : ) => x * Real.exp (-c * Real.log x ^ (1 / 10))

                            *** Prime Number Theorem (Medium Strength) *** The ChebyshevPsi function is asymptotic to x.