Datasets:

internlm

/

Lean-Github

like 21

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

have Hf : LSeriesSummable f x := by refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <| (ofReal_re x).symm ▸ hyf₁.trans_le ?_ refine (le_max_left _ (yg : EReal)).trans <| (le_max_right (x₀ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeries (f - g) ↑x = 0 x

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeries (f - g) ↑x = 0 x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

have Hg : LSeriesSummable g x := by refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <| (ofReal_re x).symm ▸ hyg₁.trans_le ?_ refine (le_max_right (yf : EReal) _).trans <| (le_max_right (x₀ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeries (f - g) ↑x = 0 x

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x Hg : LSeriesSummable g ↑x ⊢ LSeries (f - g) ↑x = 0 x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

rw [LSeries_sub Hf Hg, hx₀ x <| (le_max_left ..).trans hx, sub_self, Pi.zero_apply]

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x Hg : LSeriesSummable g ↑x ⊢ LSeries (f - g) ↑x = 0 x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <| (ofReal_re x).symm ▸ hyf₁.trans_le ?_

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeriesSummable f ↑x

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ ↑yf ≤ ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

refine (le_max_left _ (yg : EReal)).trans <| (le_max_right (x₀ : EReal) _).trans ?_

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ ↑yf ≤ ↑x

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <| (ofReal_re x).symm ▸ hyg₁.trans_le ?_

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeriesSummable g ↑x

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ ↑yg ≤ ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

refine (le_max_right (yf : EReal) _).trans <| (le_max_right (x₀ : EReal) _).trans ?_

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ ↑yg ≤ ↑x

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.eq_of_LSeries_eventually_eq

[234, 1]

[245, 74]

have hsub : (fun x : ℝ ↦ LSeries (f - g) x) =ᶠ[atTop] (0 : ℝ → ℂ) := LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq hf hg h

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 ⊢ f n = g n

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ⊢ f n = g n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.eq_of_LSeries_eventually_eq

[234, 1]

[245, 74]

have ha : abscissaOfAbsConv (f - g) ≠ ⊤ := lt_top_iff_ne_top.mp <| (abscissaOfAbsConv_sub_le f g).trans_lt <| max_lt hf hg

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ⊢ f n = g n

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ha : abscissaOfAbsConv (f - g) ≠ ⊤ ⊢ f n = g n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.eq_of_LSeries_eventually_eq

[234, 1]

[245, 74]

simpa only [Pi.sub_apply, sub_eq_zero] using (LSeries_eventually_eq_zero_iff'.mp hsub).resolve_right ha n hn

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ha : abscissaOfAbsConv (f - g) ≠ ⊤ ⊢ f n = g n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_iff_of_abscissaOfAbsConv_lt_top

[247, 1]

[254, 58]

refine eq_of_LSeries_eventually_eq hf hg ?_ hn

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ f n = g n

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] fun x => LSeries g ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_iff_of_abscissaOfAbsConv_lt_top

[247, 1]

[254, 58]

exact Filter.eventually_of_forall fun x ↦ congr_fun H x

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] fun x => LSeries g ↑x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

refine Summable.of_norm <| (hs.const_smul c).norm.of_nonneg_of_le (fun _ ↦ norm_nonneg _) fun n ↦ ?_

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c ⊢ LSeriesSummable (f * g) s

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖LSeries.term (f * g) s n‖ ≤ ‖c • LSeries.term f s n‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

rw [Complex.real_smul, ← LSeries.term_smul_apply, mul_comm]

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖LSeries.term (f * g) s n‖ ≤ ‖c • LSeries.term f s n‖

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖LSeries.term (g * f) s n‖ ≤ ‖LSeries.term (↑c • f) s n‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

refine LSeries.norm_term_le s ?_

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖LSeries.term (g * f) s n‖ ≤ ‖LSeries.term (↑c • f) s n‖

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖(g * f) n‖ ≤ ‖(↑c • f) n‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

have hc : ‖(c : ℂ)‖ = c := by simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self, (norm_nonneg _).trans (hg 0)]

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖(g * f) n‖ ≤ ‖(↑c • f) n‖

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ hc : ‖↑c‖ = c ⊢ ‖(g * f) n‖ ≤ ‖(↑c • f) n‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

simpa only [Pi.mul_apply, norm_mul, Pi.smul_apply, smul_eq_mul, hc] using mul_le_mul_of_nonneg_right (hg n) <| norm_nonneg _

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ hc : ‖↑c‖ = c ⊢ ‖(g * f) n‖ ≤ ‖(↑c • f) n‖

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_bounded

[30, 1]

[39, 61]

simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self, (norm_nonneg _).trans (hg 0)]

f g : ℕ → ℂ c : ℝ s : ℂ hs : LSeriesSummable f s hg : ∀ (n : ℕ), ‖g n‖ ≤ c n : ℕ ⊢ ‖↑c‖ = c

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_moebius

[42, 1]

[46, 36]

refine hf.mul_bounded (c := 1) fun n ↦ ?_

f : ℕ → ℂ s : ℂ hf : LSeriesSummable f s ⊢ LSeriesSummable (f * fun n => ↑(μ n)) s

f : ℕ → ℂ s : ℂ hf : LSeriesSummable f s n : ℕ ⊢ ‖↑(μ n)‖ ≤ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_moebius

[42, 1]

[46, 36]

simp only [Complex.norm_int]

f : ℕ → ℂ s : ℂ hf : LSeriesSummable f s n : ℕ ⊢ ‖↑(μ n)‖ ≤ 1

f : ℕ → ℂ s : ℂ hf : LSeriesSummable f s n : ℕ ⊢ |↑(μ n)| ≤ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeriesSummable.mul_moebius

[42, 1]

[46, 36]

exact_mod_cast abs_moebius_le_one

f : ℕ → ℂ s : ℂ hf : LSeriesSummable f s n : ℕ ⊢ |↑(μ n)| ≤ 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_convolution_distrib

[51, 1]

[59, 28]

ext n

R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R ⊢ φ * (f ⍟ g) = φ * f ⍟ (φ * g)

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ ⊢ (φ * (f ⍟ g)) n = (φ * f ⍟ (φ * g)) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_convolution_distrib

[51, 1]

[59, 28]

simp only [Pi.mul_apply, LSeries.convolution_def, Finset.mul_sum]

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ ⊢ (φ * (f ⍟ g)) n = (φ * f ⍟ (φ * g)) n

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ ⊢ ∑ i ∈ n.divisorsAntidiagonal, φ n * (f i.1 * g i.2) = ∑ x ∈ n.divisorsAntidiagonal, φ x.1 * f x.1 * (φ x.2 * g x.2)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_convolution_distrib

[51, 1]

[59, 28]

refine Finset.sum_congr rfl fun p hp ↦ ?_

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ ⊢ ∑ i ∈ n.divisorsAntidiagonal, φ n * (f i.1 * g i.2) = ∑ x ∈ n.divisorsAntidiagonal, φ x.1 * f x.1 * (φ x.2 * g x.2)

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ p : ℕ × ℕ hp : p ∈ n.divisorsAntidiagonal ⊢ φ n * (f p.1 * g p.2) = φ p.1 * f p.1 * (φ p.2 * g p.2)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_convolution_distrib

[51, 1]

[59, 28]

rw [(Nat.mem_divisorsAntidiagonal.mp hp).1.symm, hφ]

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ p : ℕ × ℕ hp : p ∈ n.divisorsAntidiagonal ⊢ φ n * (f p.1 * g p.2) = φ p.1 * f p.1 * (φ p.2 * g p.2)

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ p : ℕ × ℕ hp : p ∈ n.divisorsAntidiagonal ⊢ φ p.1 * φ p.2 * (f p.1 * g p.2) = φ p.1 * f p.1 * (φ p.2 * g p.2)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_convolution_distrib

[51, 1]

[59, 28]

exact mul_mul_mul_comm ..

case h R : Type u_1 inst✝ : CommSemiring R φ : ℕ → R hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n f g : ℕ → R n : ℕ p : ℕ × ℕ hp : p ∈ n.divisorsAntidiagonal ⊢ φ p.1 * φ p.2 * (f p.1 * g p.2) = φ p.1 * f p.1 * (φ p.2 * g p.2)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.convolution_mul_moebius

[62, 1]

[72, 68]

nth_rewrite 1 [← mul_one φ]

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n this : (1 ⍟ fun x => ↑(μ x)) = δ ⊢ φ ⍟ (φ * fun n => ↑(μ n)) = δ

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n this : (1 ⍟ fun x => ↑(μ x)) = δ ⊢ φ * 1 ⍟ (φ * fun n => ↑(μ n)) = δ

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.convolution_mul_moebius

[62, 1]

[72, 68]

simp only [← mul_convolution_distrib hφ 1 ↗μ, this, mul_delta h₁]

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n this : (1 ⍟ fun x => ↑(μ x)) = δ ⊢ φ * 1 ⍟ (φ * fun n => ↑(μ n)) = δ

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.convolution_mul_moebius

[62, 1]

[72, 68]

rw [one_convolution_eq_zeta_convolution, ← one_eq_delta]

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n ⊢ (1 ⍟ fun x => ↑(μ x)) = δ

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n ⊢ ((fun x => ↑(ζ x)) ⍟ fun x => ↑(μ x)) = fun n => 1 n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.convolution_mul_moebius

[62, 1]

[72, 68]

change ⇑(ζ : ArithmeticFunction ℂ) ⍟ ⇑(μ : ArithmeticFunction ℂ) = ⇑(1 : ArithmeticFunction ℂ)

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n ⊢ ((fun x => ↑(ζ x)) ⍟ fun x => ↑(μ x)) = fun n => 1 n

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n ⊢ ⇑↑ζ ⍟ ⇑↑μ = ⇑1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.convolution_mul_moebius

[62, 1]

[72, 68]

simp only [coe_mul, coe_zeta_mul_coe_moebius]

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n ⊢ ⇑↑ζ ⍟ ⇑↑μ = ⇑1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_mu_eq_one

[75, 1]

[80, 23]

rw [← LSeries_convolution' hs ?_, convolution_mul_moebius h₁ hφ, LSeries_delta, Pi.one_apply]

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n s : ℂ hs : LSeriesSummable φ s ⊢ L φ s * L (φ * fun n => ↑(μ n)) s = 1

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n s : ℂ hs : LSeriesSummable φ s ⊢ LSeriesSummable (φ * fun n => ↑(μ n)) s

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

LSeries.mul_mu_eq_one

[75, 1]

[80, 23]

exact hs.mul_moebius

φ : ℕ → ℂ h₁ : φ 1 = 1 hφ : ∀ (m n : ℕ), φ (m * n) = φ m * φ n s : ℂ hs : LSeriesSummable φ s ⊢ LSeriesSummable (φ * fun n => ↑(μ n)) s

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

DirichletCharacter.toFun_on_nat_map_one

[92, 1]

[93, 32]

simp only [cast_one, map_one]

N : ℕ χ : DirichletCharacter ℂ N ⊢ (fun n => χ ↑n) 1 = 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/DirichletLSeries.lean

DirichletCharacter.toFun_on_nat_map_mul

[95, 1]

[97, 32]

simp only [cast_mul, map_mul]

N : ℕ χ : DirichletCharacter ℂ N m n : ℕ ⊢ (fun n => χ ↑n) (m * n) = (fun n => χ ↑n) m * (fun n => χ ↑n) n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

rw [LSeries]

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => χ ↑n) s

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

convert exp_sum_primes_log_eq_tsum (f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs) <| summable_dirichletSummand χ hs

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n

case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

ext n

case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)

case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

rcases eq_or_ne n 0 with rfl | hn

case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0 case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

simp only [term_zero, map_zero]

case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

DirichletCharacter.LSeries_eulerProduct'

[42, 1]

[51, 61]

simp [hn, dirichletSummandHom, div_eq_mul_inv, cpow_neg]

case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

ArithmeticFunction.LSeries_zeta_eulerProduct'

[56, 1]

[59, 62]

convert modOne_eq_one (R := ℂ) ▸ LSeries_eulerProduct' χ₁ hs using 7

s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - ↑↑p ^ (-s)).log) = L 1 s

case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

ArithmeticFunction.LSeries_zeta_eulerProduct'

[56, 1]

[59, 62]

rw [MulChar.one_apply <| isUnit_of_subsingleton _, one_mul]

case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

have (p : Nat.Primes) : ‖χ p * (p : ℂ) ^ (-s)‖ ≤ (p : ℝ) ^ (-s).re := by rw [norm_mul, norm_natCast_cpow_of_re_ne_zero _ <| re_neg_ne_zero_of_one_lt_re hs] calc ‖χ p‖ * (p : ℝ) ^ (-s).re _ ≤ 1 * (p : ℝ) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one χ _ _ = _ := one_mul _

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ Summable fun p => -(1 - χ ↑↑p * ↑↑p ^ (-s)).log

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re this : ∀ (p : Nat.Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re ⊢ Summable fun p => -(1 - χ ↑↑p * ↑↑p ^ (-s)).log

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

refine (Nat.Primes.summable_rpow.mpr ?_).of_nonneg_of_le (fun _ ↦ norm_nonneg _) this |>.of_norm.neg_clog_one_sub

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re this : ∀ (p : Nat.Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re ⊢ Summable fun p => -(1 - χ ↑↑p * ↑↑p ^ (-s)).log

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re this : ∀ (p : Nat.Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re ⊢ (-s).re < -1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

simp only [neg_re, neg_lt_neg_iff, hs]

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re this : ∀ (p : Nat.Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re ⊢ (-s).re < -1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

rw [norm_mul, norm_natCast_cpow_of_re_ne_zero _ <| re_neg_ne_zero_of_one_lt_re hs]

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p‖ * ↑↑p ^ (-s).re ≤ ↑↑p ^ (-s).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

calc ‖χ p‖ * (p : ℝ) ^ (-s).re _ ≤ 1 * (p : ℝ) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one χ _ _ = _ := one_mul _

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p‖ * ↑↑p ^ (-s).re ≤ ↑↑p ^ (-s).re

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

gcongr

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p‖ * ↑↑p ^ (-s).re ≤ 1 * ↑↑p ^ (-s.re)

case h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p‖ ≤ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

summable_neg_log_one_sub_char_mul_prime_cpow

[69, 1]

[79, 41]

exact DirichletCharacter.norm_le_one χ _

case h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re p : Nat.Primes ⊢ ‖χ ↑↑p‖ ≤ 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have hac₀ : ‖(a : ℂ)‖ < 1 := by simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg ha₀, ha₁]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have hac₁ : ‖a * z‖ < 1 := by rwa [norm_mul, hz, mul_one]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have hac₂ : ‖a * z ^ 2‖ < 1 := by rwa [norm_mul, norm_pow, hz, one_pow, mul_one]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have H₀ := (hasSum_re <| hasSum_taylorSeries_neg_log hac₀).mul_left 3

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have H₁ := (hasSum_re <| hasSum_taylorSeries_neg_log hac₁).mul_left 4

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

have H₂ := hasSum_re <| hasSum_taylorSeries_neg_log hac₂

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) H₂ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

rw [← ((H₀.add H₁).add H₂).tsum_eq]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) H₂ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊢ 0 ≤ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) H₂ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊢ 0 ≤ ∑' (b : ℕ), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

clear H₀ H₁ H₂

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 H₀ : HasSum (fun i => 3 * (↑a ^ i / ↑i).re) (3 * (-(1 - ↑a).log).re) H₁ : HasSum (fun i => 4 * ((↑a * z) ^ i / ↑i).re) (4 * (-(1 - ↑a * z).log).re) H₂ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊢ 0 ≤ ∑' (b : ℕ), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ ∑' (b : ℕ), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

refine tsum_nonneg fun n ↦ ?_

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ ∑' (b : ℕ), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ ⊢ 0 ≤ 3 * (↑a ^ n / ↑n).re + 4 * ((↑a * z) ^ n / ↑n).re + ((↑a * z ^ 2) ^ n / ↑n).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

simp only [mul_pow, ← ofReal_pow, div_natCast_re, ofReal_re, mul_re, ofReal_im, zero_mul, sub_zero]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ ⊢ 0 ≤ 3 * (↑a ^ n / ↑n).re + 4 * ((↑a * z) ^ n / ↑n).re + ((↑a * z ^ 2) ^ n / ↑n).re

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ ⊢ 0 ≤ 3 * (a ^ n / ↑n) + 4 * (a ^ n * (z ^ n).re / ↑n) + a ^ n * ((z ^ 2) ^ n).re / ↑n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

rcases n.eq_zero_or_pos with rfl | hn

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ ⊢ 0 ≤ 3 * (a ^ n / ↑n) + 4 * (a ^ n * (z ^ n).re / ↑n) + a ^ n * ((z ^ 2) ^ n).re / ↑n

case inl a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ 3 * (a ^ 0 / ↑0) + 4 * (a ^ 0 * (z ^ 0).re / ↑0) + a ^ 0 * ((z ^ 2) ^ 0).re / ↑0 case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * (a ^ n / ↑n) + 4 * (a ^ n * (z ^ n).re / ↑n) + a ^ n * ((z ^ 2) ^ n).re / ↑n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

field_simp

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * (a ^ n / ↑n) + 4 * (a ^ n * (z ^ n).re / ↑n) + a ^ n * ((z ^ 2) ^ n).re / ↑n

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / ↑n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

refine div_nonneg ?_ n.cast_nonneg

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / ↑n

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

rw [← pow_mul, pow_mul', sq, mul_re, ← sq, ← sq, ← sq_abs_sub_sq_re, ← norm_eq_abs, norm_pow, hz]

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

calc 0 ≤ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 := by positivity _ = _ := by ring

case inr a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg ha₀, ha₁]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 ⊢ ‖↑a‖ < 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

rwa [norm_mul, hz, mul_one]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 ⊢ ‖↑a * z‖ < 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

rwa [norm_mul, norm_pow, hz, one_pow, mul_one]

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 ⊢ ‖↑a * z ^ 2‖ < 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

simp

case inl a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 ⊢ 0 ≤ 3 * (a ^ 0 / ↑0) + 4 * (a ^ 0 * (z ^ 0).re / ↑0) + a ^ 0 * ((z ^ 2) ^ 0).re / ↑0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

positivity

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 0 ≤ 2 * a ^ n * ((z ^ n).re + 1) ^ 2

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg'

[81, 1]

[104, 22]

ring

a : ℝ ha₀ : 0 ≤ a ha₁ : a < 1 z : ℂ hz : ‖z‖ = 1 hac₀ : ‖↑a‖ < 1 hac₁ : ‖↑a * z‖ < 1 hac₂ : ‖↑a * z ^ 2‖ < 1 n : ℕ hn : n > 0 ⊢ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 = 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

by_cases hn' : IsUnit (n : ZMod N)

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re case neg N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : ¬IsUnit ↑n ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

have ha₀ : 0 ≤ (n : ℝ) ^ (-x) := Real.rpow_nonneg n.cast_nonneg _

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

have ha₁ : (n : ℝ) ^ (-x) < 1 := by simpa only [Real.rpow_lt_one_iff n.cast_nonneg, Nat.cast_eq_zero, Nat.one_lt_cast, Left.neg_neg_iff, Nat.cast_lt_one, Left.neg_pos_iff] using Or.inr <| Or.inl ⟨hn, zero_lt_one.trans hx⟩

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

have hz : ‖χ n * (n : ℂ) ^ (-(I * y))‖ = 1 := by rw [norm_mul, ← hn'.unit_spec, DirichletCharacter.unit_norm_eq_one χ hn'.unit, one_mul, norm_eq_abs, abs_cpow_of_imp fun h ↦ False.elim <| by linarith [Nat.cast_eq_zero.mp h, hn]] simp

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

rw [MulChar.one_apply hn', one_mul]

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ 0 ≤ 3 * (-(1 - ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

convert re_log_comb_nonneg' ha₀ ha₁ hz using 6

case pos N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ 0 ≤ 3 * (-(1 - ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ (1 - ↑n ^ (-↑x)).log = (1 - ↑(↑n ^ (-x))).log case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log = (1 - ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y)))).log case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

simpa only [Real.rpow_lt_one_iff n.cast_nonneg, Nat.cast_eq_zero, Nat.one_lt_cast, Left.neg_neg_iff, Nat.cast_lt_one, Left.neg_pos_iff] using Or.inr <| Or.inl ⟨hn, zero_lt_one.trans hx⟩

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ⊢ ↑n ^ (-x) < 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

rw [norm_mul, ← hn'.unit_spec, DirichletCharacter.unit_norm_eq_one χ hn'.unit, one_mul, norm_eq_abs, abs_cpow_of_imp fun h ↦ False.elim <| by linarith [Nat.cast_eq_zero.mp h, hn]]

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊢ ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊢ Complex.abs ↑n ^ (-(I * ↑y)).re / ((↑n).arg * (-(I * ↑y)).im).exp = 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

simp

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊢ Complex.abs ↑n ^ (-(I * ↑y)).re / ((↑n).arg * (-(I * ↑y)).im).exp = 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

linarith [Nat.cast_eq_zero.mp h, hn]

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 h : ↑n = 0 ⊢ False

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

congr 2

case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ (1 - ↑n ^ (-↑x)).log = (1 - ↑(↑n ^ (-x))).log

case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ ↑n ^ (-↑x) = ↑(↑n ^ (-x))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

exact_mod_cast (ofReal_cpow n.cast_nonneg (-x)).symm

case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ ↑n ^ (-↑x) = ↑(↑n ^ (-x))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

congr 2

case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log = (1 - ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y)))).log

case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n * ↑n ^ (-(↑x + I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y)))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

rw [neg_add, cpow_add _ _ <| by norm_cast; linarith, ← ofReal_neg, ofReal_cpow n.cast_nonneg (-x), ofReal_natCast]

case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n * ↑n ^ (-(↑x + I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y)))

case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n * (↑n ^ ↑(-x) * ↑n ^ (-(I * ↑y))) = ↑n ^ ↑(-x) * (χ ↑n * ↑n ^ (-(I * ↑y)))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

ring

case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n * (↑n ^ ↑(-x) * ↑n ^ (-(I * ↑y))) = ↑n ^ ↑(-x) * (χ ↑n * ↑n ^ (-(I * ↑y)))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

norm_cast

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ ↑n ≠ 0

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ ¬n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

linarith

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ ¬n = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

rw [neg_add, cpow_add _ _ <| by norm_cast; linarith, ← ofReal_neg, ofReal_cpow n.cast_nonneg (-x), ofReal_natCast, show -(2 * I * y) = (2 : ℕ) * (-I * y) by ring, cpow_nat_mul]

case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n ^ 2 * (↑n ^ ↑(-x) * (↑n ^ (-I * ↑y)) ^ 2) = ↑n ^ ↑(-x) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

ring_nf

case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6 N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ χ ↑n ^ 2 * (↑n ^ ↑(-x) * (↑n ^ (-I * ↑y)) ^ 2) = ↑n ^ ↑(-x) * (χ ↑n * ↑n ^ (-(I * ↑y))) ^ 2

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

ring

N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ha₀ : 0 ≤ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1 ⊢ -(2 * I * ↑y) = ↑2 * (-I * ↑y)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

re_log_comb_nonneg_dirichlet

[106, 1]

[135, 37]

simp [MulChar.map_nonunit _ hn']

case neg N : ℕ χ : DirichletCharacter ℂ N n : ℕ hn : 2 ≤ n x y : ℝ hx : 1 < x hn' : ¬IsUnit ↑n ⊢ 0 ≤ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

one_lt_re_of_pos

[138, 1]

[141, 92]

simp only [add_re, one_re, ofReal_re, lt_add_iff_pos_right, hx, mul_re, I_re, zero_mul, I_im, ofReal_im, mul_zero, sub_self, add_zero, re_ofNat, im_ofNat, mul_one, mul_im, and_self]

x y : ℝ hx : 0 < x ⊢ 1 < (1 + ↑x).re ∧ 1 < (1 + ↑x + I * ↑y).re ∧ 1 < (1 + ↑x + 2 * I * ↑y).re

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

norm_dirichlet_product_ge_one

[147, 1]

[174, 33]

let χ₀ := (1 : DirichletCharacter ℂ N)

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

norm_dirichlet_product_ge_one

[147, 1]

[174, 33]

have ⟨h₀, h₁, h₂⟩ := one_lt_re_of_pos y hx

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

norm_dirichlet_product_ge_one

[147, 1]

[174, 33]

have hx₁ : 1 + (x : ℂ) = (1 + x : ℂ).re := by simp only [add_re, one_re, ofReal_re, ofReal_add, ofReal_one]

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

norm_dirichlet_product_ge_one

[147, 1]

[174, 33]

have hsum₀ := (hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow χ₀ h₀).hasSum).summable.mul_left 3

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsum₀ : Summable fun i => 3 * (-(1 - χ₀ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/PNT.lean

norm_dirichlet_product_ge_one

[147, 1]

[174, 33]

have hsum₁ := (hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow χ h₁).hasSum).summable.mul_left 4

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsum₀ : Summable fun i => 3 * (-(1 - χ₀ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1

N : ℕ χ : DirichletCharacter ℂ N x : ℝ hx : 0 < x y : ℝ χ₀ : DirichletCharacter ℂ N := 1 h₀ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsum₀ : Summable fun i => 3 * (-(1 - χ₀ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re hsum₁ : Summable fun i => 4 * (-(1 - χ ↑↑i * ↑↑i ^ (-(1 + ↑x + I * ↑y))).log).re ⊢ ‖L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => χ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (χ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)‖ ≥ 1