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2.09M
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 β–Έ hyf₁.trans_le ?_ refine (le_max_left _ (yg : EReal)).trans <| (le_max_right (xβ‚€ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ LSeries (f - g) ↑x = 0 x
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 β–Έ hyg₁.trans_le ?_ refine (le_max_right (yf : EReal) _).trans <| (le_max_right (xβ‚€ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ LSeries (f - g) ↑x = 0 x
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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, hxβ‚€ x <| (le_max_left ..).trans hx, sub_self, Pi.zero_apply]
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 β–Έ hyf₁.trans_le ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ LSeriesSummable f ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 (xβ‚€ : EReal) _).trans ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ ↑yf ≀ ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ max (↑xβ‚€) (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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ max (↑xβ‚€) (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 β–Έ hyg₁.trans_le ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ LSeriesSummable g ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 (xβ‚€ : EReal) _).trans ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ ↑yg ≀ ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ max (↑xβ‚€) (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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ max (↑xβ‚€) (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) =αΆ [atTop] (0 : ℝ β†’ β„‚) := LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq hf hg h
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [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) β‰  ⊀ := lt_top_iff_ne_top.mp <| (abscissaOfAbsConv_sub_le f g).trans_lt <| max_lt hf hg
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ha : abscissaOfAbsConv (f - g) β‰  ⊀ ⊒ 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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ha : abscissaOfAbsConv (f - g) β‰  ⊀ ⊒ 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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ (fun x => LSeries f ↑x) =αΆ [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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ (fun x => LSeries f ↑x) =αΆ [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 : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c ⊒ LSeriesSummable (f * g) s
f g : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–LSeries.term (f * g) s nβ€– ≀ β€–c β€’ LSeries.term f s nβ€–
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 : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–LSeries.term (f * g) s nβ€– ≀ β€–c β€’ LSeries.term f s nβ€–
f g : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–LSeries.term (g * f) s nβ€– ≀ β€–LSeries.term (↑c β€’ f) s nβ€–
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 : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–LSeries.term (g * f) s nβ€– ≀ β€–LSeries.term (↑c β€’ f) s nβ€–
f g : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–(g * f) nβ€– ≀ β€–(↑c β€’ f) nβ€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/DirichletLSeries.lean
LSeriesSummable.mul_bounded
[30, 1]
[39, 61]
have hc : β€–(c : β„‚)β€– = c := by simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self, (norm_nonneg _).trans (hg 0)]
f g : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ β€–(g * f) nβ€– ≀ β€–(↑c β€’ f) nβ€–
f g : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• hc : ‖↑cβ€– = c ⊒ β€–(g * f) nβ€– ≀ β€–(↑c β€’ f) nβ€–
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 : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• hc : ‖↑cβ€– = c ⊒ β€–(g * f) nβ€– ≀ β€–(↑c β€’ f) nβ€–
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 : β„• β†’ β„‚ c : ℝ s : β„‚ hs : LSeriesSummable f s hg : βˆ€ (n : β„•), β€–g nβ€– ≀ c n : β„• ⊒ ‖↑cβ€– = 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 : β„• β†’ β„‚ s : β„‚ hf : LSeriesSummable f s ⊒ LSeriesSummable (f * fun n => ↑(ΞΌ n)) s
f : β„• β†’ β„‚ s : β„‚ hf : LSeriesSummable f s n : β„• ⊒ ‖↑(ΞΌ n)β€– ≀ 1
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/DirichletLSeries.lean
LSeriesSummable.mul_moebius
[42, 1]
[46, 36]
simp only [Complex.norm_int]
f : β„• β†’ β„‚ s : β„‚ hf : LSeriesSummable f s n : β„• ⊒ ‖↑(ΞΌ n)β€– ≀ 1
f : β„• β†’ β„‚ s : β„‚ hf : LSeriesSummable f s n : β„• ⊒ |↑(ΞΌ 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 : β„• β†’ β„‚ s : β„‚ hf : LSeriesSummable f s n : β„• ⊒ |↑(ΞΌ 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 Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R ⊒ Ο† * (f ⍟ g) = Ο† * f ⍟ (Ο† * g)
case h R : Type u_1 inst✝ : CommSemiring R Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• ⊒ (Ο† * (f ⍟ g)) n = (Ο† * f ⍟ (Ο† * 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 Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• ⊒ (Ο† * (f ⍟ g)) n = (Ο† * f ⍟ (Ο† * g)) n
case h R : Type u_1 inst✝ : CommSemiring R Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• ⊒ βˆ‘ i ∈ n.divisorsAntidiagonal, Ο† n * (f i.1 * g i.2) = βˆ‘ x ∈ n.divisorsAntidiagonal, Ο† x.1 * f x.1 * (Ο† 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 Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• ⊒ βˆ‘ i ∈ n.divisorsAntidiagonal, Ο† n * (f i.1 * g i.2) = βˆ‘ x ∈ n.divisorsAntidiagonal, Ο† x.1 * f x.1 * (Ο† x.2 * g x.2)
case h R : Type u_1 inst✝ : CommSemiring R Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• p : β„• Γ— β„• hp : p ∈ n.divisorsAntidiagonal ⊒ Ο† n * (f p.1 * g p.2) = Ο† p.1 * f p.1 * (Ο† 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 Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• p : β„• Γ— β„• hp : p ∈ n.divisorsAntidiagonal ⊒ Ο† n * (f p.1 * g p.2) = Ο† p.1 * f p.1 * (Ο† p.2 * g p.2)
case h R : Type u_1 inst✝ : CommSemiring R Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• p : β„• Γ— β„• hp : p ∈ n.divisorsAntidiagonal ⊒ Ο† p.1 * Ο† p.2 * (f p.1 * g p.2) = Ο† p.1 * f p.1 * (Ο† 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 Ο† : β„• β†’ R hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n f g : β„• β†’ R n : β„• p : β„• Γ— β„• hp : p ∈ n.divisorsAntidiagonal ⊒ Ο† p.1 * Ο† p.2 * (f p.1 * g p.2) = Ο† p.1 * f p.1 * (Ο† 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 Ο†]
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n this : (1 ⍟ fun x => ↑(ΞΌ x)) = Ξ΄ ⊒ Ο† ⍟ (Ο† * fun n => ↑(ΞΌ n)) = Ξ΄
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n this : (1 ⍟ fun x => ↑(ΞΌ x)) = Ξ΄ ⊒ Ο† * 1 ⍟ (Ο† * fun n => ↑(ΞΌ n)) = Ξ΄
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/DirichletLSeries.lean
LSeries.convolution_mul_moebius
[62, 1]
[72, 68]
simp only [← mul_convolution_distrib hΟ† 1 β†—ΞΌ, this, mul_delta h₁]
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n this : (1 ⍟ fun x => ↑(ΞΌ x)) = Ξ΄ ⊒ Ο† * 1 ⍟ (Ο† * fun n => ↑(ΞΌ n)) = Ξ΄
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]
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n ⊒ (1 ⍟ fun x => ↑(ΞΌ x)) = Ξ΄
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n ⊒ ((fun x => ↑(ΞΆ x)) ⍟ fun x => ↑(ΞΌ x)) = fun n => 1 n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/DirichletLSeries.lean
LSeries.convolution_mul_moebius
[62, 1]
[72, 68]
change ⇑(ΞΆ : ArithmeticFunction β„‚) ⍟ ⇑(ΞΌ : ArithmeticFunction β„‚) = ⇑(1 : ArithmeticFunction β„‚)
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n ⊒ ((fun x => ↑(ΞΆ x)) ⍟ fun x => ↑(ΞΌ x)) = fun n => 1 n
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n ⊒ ⇑↑΢ ⍟ ⇑↑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]
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n ⊒ ⇑↑΢ ⍟ ⇑↑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 hΟ†, LSeries_delta, Pi.one_apply]
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n s : β„‚ hs : LSeriesSummable Ο† s ⊒ L Ο† s * L (Ο† * fun n => ↑(ΞΌ n)) s = 1
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n s : β„‚ hs : LSeriesSummable Ο† s ⊒ LSeriesSummable (Ο† * fun n => ↑(ΞΌ 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
Ο† : β„• β†’ β„‚ h₁ : Ο† 1 = 1 hΟ† : βˆ€ (m n : β„•), Ο† (m * n) = Ο† m * Ο† n s : β„‚ hs : LSeriesSummable Ο† s ⊒ LSeriesSummable (Ο† * fun n => ↑(ΞΌ 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 : β„• Ο‡ : DirichletCharacter β„‚ N ⊒ (fun n => Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N m n : β„• ⊒ (fun n => Ο‡ ↑n) (m * n) = (fun n => Ο‡ ↑n) m * (fun n => Ο‡ ↑n) n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
DirichletCharacter.LSeries_eulerProduct'
[42, 1]
[51, 61]
rw [LSeries]
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => Ο‡ ↑n) s
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = βˆ‘' (n : β„•), term (fun n => Ο‡ ↑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 Ο‡ <| ne_zero_of_one_lt_re hs) <| summable_dirichletSummand Ο‡ hs
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = βˆ‘' (n : β„•), term (fun n => Ο‡ ↑n) s n
case h.e'_3.h.e'_5.h.h.e N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s = ⇑(dirichletSummandHom Ο‡ β‹―)
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s = ⇑(dirichletSummandHom Ο‡ β‹―)
case h.e'_3.h.e'_5.h.h.e.h N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) 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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n
case h.e'_3.h.e'_5.h.h.e.h.inl N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s 0 = (dirichletSummandHom Ο‡ β‹―) 0 case h.e'_3.h.e'_5.h.h.e.h.inr N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• hn : n β‰  0 ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) 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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s 0 = (dirichletSummandHom Ο‡ β‹―) 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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• hn : n β‰  0 ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) 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 := β„‚) β–Έ LSeries_eulerProduct' χ₁ hs using 7
s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (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 : β„‚ 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 : β„‚ 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) : β€–Ο‡ p * (p : β„‚) ^ (-s)β€– ≀ (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 β€–Ο‡ pβ€– * (p : ℝ) ^ (-s).re _ ≀ 1 * (p : ℝ) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one Ο‡ _ _ = _ := one_mul _
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ Summable fun p => -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re this : βˆ€ (p : Nat.Primes), β€–Ο‡ ↑↑p * ↑↑p ^ (-s)β€– ≀ ↑↑p ^ (-s).re ⊒ Summable fun p => -(1 - Ο‡ ↑↑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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re this : βˆ€ (p : Nat.Primes), β€–Ο‡ ↑↑p * ↑↑p ^ (-s)β€– ≀ ↑↑p ^ (-s).re ⊒ Summable fun p => -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re this : βˆ€ (p : Nat.Primes), β€–Ο‡ ↑↑p * ↑↑p ^ (-s)β€– ≀ ↑↑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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re this : βˆ€ (p : Nat.Primes), β€–Ο‡ ↑↑p * ↑↑p ^ (-s)β€– ≀ ↑↑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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑p * ↑↑p ^ (-s)β€– ≀ ↑↑p ^ (-s).re
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑pβ€– * ↑↑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 β€–Ο‡ pβ€– * (p : ℝ) ^ (-s).re _ ≀ 1 * (p : ℝ) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one Ο‡ _ _ = _ := one_mul _
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑pβ€– * ↑↑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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑pβ€– * ↑↑p ^ (-s).re ≀ 1 * ↑↑p ^ (-s.re)
case h N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑pβ€– ≀ 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 Ο‡ _
case h N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re p : Nat.Primes ⊒ β€–Ο‡ ↑↑pβ€– ≀ 1
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
re_log_comb_nonneg'
[81, 1]
[104, 22]
have hacβ‚€ : β€–(a : β„‚)β€– < 1 := by simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg haβ‚€, ha₁]
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 ⊒ 0 ≀ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 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 : β€–a * zβ€– < 1 := by rwa [norm_mul, hz, mul_one]
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 ⊒ 0 ≀ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 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 hacβ‚‚ : β€–a * z ^ 2β€– < 1 := by rwa [norm_mul, norm_pow, hz, one_pow, mul_one]
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 ⊒ 0 ≀ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 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 Hβ‚€ := (hasSum_re <| hasSum_taylorSeries_neg_log hacβ‚€).mul_left 3
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 ⊒ 0 ≀ 3 * (-(1 - ↑a).log).re + 4 * (-(1 - ↑a * z).log).re + (-(1 - ↑a * z ^ 2).log).re
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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 Hβ‚‚ := hasSum_re <| hasSum_taylorSeries_neg_log hacβ‚‚
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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) Hβ‚‚ : 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 [← ((Hβ‚€.add H₁).add Hβ‚‚).tsum_eq]
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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) Hβ‚‚ : 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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) Hβ‚‚ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊒ 0 ≀ βˆ‘' (b : β„•), (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 Hβ‚€ H₁ Hβ‚‚
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 Hβ‚€ : 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) Hβ‚‚ : HasSum (fun x => ((↑a * z ^ 2) ^ x / ↑x).re) (-(1 - ↑a * z ^ 2).log).re ⊒ 0 ≀ βˆ‘' (b : β„•), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 ⊒ 0 ≀ βˆ‘' (b : β„•), (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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 ⊒ 0 ≀ βˆ‘' (b : β„•), (3 * (↑a ^ b / ↑b).re + 4 * ((↑a * z) ^ b / ↑b).re + ((↑a * z ^ 2) ^ b / ↑b).re)
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• ⊒ 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• ⊒ 0 ≀ 3 * (↑a ^ n / ↑n).re + 4 * ((↑a * z) ^ n / ↑n).re + ((↑a * z ^ 2) ^ n / ↑n).re
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• ⊒ 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• ⊒ 0 ≀ 3 * (a ^ n / ↑n) + 4 * (a ^ n * (z ^ n).re / ↑n) + a ^ n * ((z ^ 2) ^ n).re / ↑n
case inl a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 ⊒ 0 ≀ 3 * (a ^ 0 / ↑0) + 4 * (a ^ 0 * (z ^ 0).re / ↑0) + a ^ 0 * ((z ^ 2) ^ 0).re / ↑0 case inr a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• hn : n > 0 ⊒ 0 ≀ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / ↑n
case inr a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• hn : n > 0 ⊒ 0 ≀ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re
case inr a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€, ha₁]
a : ℝ haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 ⊒ ‖↑aβ€– < 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 ⊒ ‖↑a * zβ€– < 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 ⊒ ‖↑a * z ^ 2β€– < 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 haβ‚€ : 0 ≀ a ha₁ : a < 1 z : β„‚ hz : β€–zβ€– = 1 hacβ‚€ : ‖↑aβ€– < 1 hac₁ : ‖↑a * zβ€– < 1 hacβ‚‚ : ‖↑a * z ^ 2β€– < 1 n : β„• 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re case neg N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : Β¬IsUnit ↑n ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 haβ‚€ : 0 ≀ (n : ℝ) ^ (-x) := Real.rpow_nonneg n.cast_nonneg _
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 : β€–Ο‡ n * (n : β„‚) ^ (-(I * y))β€– = 1 := by rw [norm_mul, ← hn'.unit_spec, DirichletCharacter.unit_norm_eq_one Ο‡ 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))).log).re
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ 0 ≀ 3 * (-(1 - ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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' haβ‚€ ha₁ hz using 6
case pos N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ 0 ≀ 3 * (-(1 - ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ (1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log = (1 - ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 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 Ο‡ 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 ⊒ β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1
N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ (1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log = (1 - ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y)) = ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y)) = ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n * (↑n ^ ↑(-x) * ↑n ^ (-(I * ↑y))) = ↑n ^ ↑(-x) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n * (↑n ^ ↑(-x) * ↑n ^ (-(I * ↑y))) = ↑n ^ ↑(-x) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ ↑n β‰  0
N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Β¬n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
re_log_comb_nonneg_dirichlet
[106, 1]
[135, 37]
linarith
N : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Β¬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 : β„•) * (-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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y)) = ↑(↑n ^ (-x)) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n ^ 2 * (↑n ^ ↑(-x) * (↑n ^ (-I * ↑y)) ^ 2) = ↑n ^ ↑(-x) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 1 ⊒ Ο‡ ↑n ^ 2 * (↑n ^ ↑(-x) * (↑n ^ (-I * ↑y)) ^ 2) = ↑n ^ ↑(-x) * (Ο‡ ↑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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : IsUnit ↑n haβ‚€ : 0 ≀ ↑n ^ (-x) ha₁ : ↑n ^ (-x) < 1 hz : β€–Ο‡ ↑n * ↑n ^ (-(I * ↑y))β€– = 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 : β„• Ο‡ : DirichletCharacter β„‚ N n : β„• hn : 2 ≀ n x y : ℝ hx : 1 < x hn' : Β¬IsUnit ↑n ⊒ 0 ≀ 3 * (-(1 - 1 ↑n * ↑n ^ (-↑x)).log).re + 4 * (-(1 - Ο‡ ↑n * ↑n ^ (-(↑x + I * ↑y))).log).re + (-(1 - Ο‡ ↑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 Ο‡β‚€ := (1 : DirichletCharacter β„‚ N)
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
norm_dirichlet_product_ge_one
[147, 1]
[174, 33]
have ⟨hβ‚€, h₁, hβ‚‚βŸ© := one_lt_re_of_pos y hx
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
norm_dirichlet_product_ge_one
[147, 1]
[174, 33]
have hx₁ : 1 + (x : β„‚) = (1 + x : β„‚).re := by simp only [add_re, one_re, ofReal_re, ofReal_add, ofReal_one]
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/PNT.lean
norm_dirichlet_product_ge_one
[147, 1]
[174, 33]
have hsumβ‚€ := (hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow Ο‡β‚€ hβ‚€).hasSum).summable.mul_left 3
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsumβ‚€ : Summable fun i => 3 * (-(1 - Ο‡β‚€ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 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 Ο‡ h₁).hasSum).summable.mul_left 4
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsumβ‚€ : Summable fun i => 3 * (-(1 - Ο‡β‚€ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1
N : β„• Ο‡ : DirichletCharacter β„‚ N x : ℝ hx : 0 < x y : ℝ Ο‡β‚€ : DirichletCharacter β„‚ N := 1 hβ‚€ : 1 < (1 + ↑x).re h₁ : 1 < (1 + ↑x + I * ↑y).re hβ‚‚ : 1 < (1 + ↑x + 2 * I * ↑y).re hx₁ : 1 + ↑x = ↑(1 + ↑x).re hsumβ‚€ : Summable fun i => 3 * (-(1 - Ο‡β‚€ ↑↑i * ↑↑i ^ (-(1 + ↑x))).log).re hsum₁ : Summable fun i => 4 * (-(1 - Ο‡ ↑↑i * ↑↑i ^ (-(1 + ↑x + I * ↑y))).log).re ⊒ β€–L (fun n => 1 ↑n) (1 + ↑x) ^ 3 * L (fun n => Ο‡ ↑n) (1 + ↑x + I * ↑y) ^ 4 * L (fun n => (Ο‡ ^ 2) ↑n) (1 + ↑x + 2 * I * ↑y)β€– β‰₯ 1