url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct
|
[94, 1]
|
[99, 87]
|
refine Tendsto.congr (fun n ↦ Finset.prod_congr rfl fun p hp ↦ ?_) <|
eulerProduct_of_completelyMultiplicative (toFun_on_nat_map_one χ) (toFun_on_nat_map_mul χ) <|
LSeriesSummable_of_one_lt_re χ hs
|
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - χ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n => χ ↑n) s))
|
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
n p : ℕ
hp : p ∈ n.primesBelow
⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct
|
[94, 1]
|
[99, 87]
|
rw [term_of_ne_zero (prime_of_mem_primesBelow hp).ne_zero, cpow_neg, div_eq_mul_inv]
|
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
n p : ℕ
hp : p ∈ n.primesBelow
⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
DirichletCharacter.LSeries_eulerProduct'
|
[102, 1]
|
[110, 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/EulerProduct.lean
|
ArithmeticFunction.LSeries_zeta_eulerProduct'
|
[117, 1]
|
[120, 62]
|
convert modOne_eq_one (R := ℂ) ▸ LSeries_eulerProduct' (1 : DirichletCharacter ℂ 1) 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/EulerProduct.lean
|
ArithmeticFunction.LSeries_zeta_eulerProduct'
|
[117, 1]
|
[120, 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/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
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
have hsum₂ :=
(hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow (χ ^ 2) h₂).hasSum).summable
|
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
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * 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
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
rw [← DirichletCharacter.LSeries_eulerProduct' _ h₀,
← DirichletCharacter.LSeries_eulerProduct' χ h₁,
← DirichletCharacter.LSeries_eulerProduct' (χ ^ 2) h₂, ← exp_nat_mul, ← exp_nat_mul, ← exp_add,
← exp_add, norm_eq_abs, abs_exp]
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * 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
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
⊢ (↑3 * ∑' (p : Primes), -(1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x))).log +
↑4 * ∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y))).log +
∑' (p : Primes), -(1 - (χ ^ 2) ↑↑p * ↑↑p ^ (-(1 + ↑x + 2 * I * ↑y))).log).re.exp ≥
1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
simp only [Nat.cast_ofNat, add_re, mul_re, re_ofNat, im_ofNat, zero_mul, sub_zero,
Real.one_le_exp_iff]
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
⊢ (↑3 * ∑' (p : Primes), -(1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x))).log +
↑4 * ∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y))).log +
∑' (p : Primes), -(1 - (χ ^ 2) ↑↑p * ↑↑p ^ (-(1 + ↑x + 2 * I * ↑y))).log).re.exp ≥
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
⊢ 0 ≤
3 * (∑' (p : Primes), -(1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x))).log).re +
4 * (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y))).log).re +
(∑' (p : Primes), -(1 - (χ ^ 2) ↑↑p * ↑↑p ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
rw [re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ h₀,
re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ h₁,
re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ h₂, ← tsum_mul_left, ← tsum_mul_left,
← tsum_add hsum₀ hsum₁, ← tsum_add (hsum₀.add hsum₁) hsum₂]
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
⊢ 0 ≤
3 * (∑' (p : Primes), -(1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x))).log).re +
4 * (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y))).log).re +
(∑' (p : Primes), -(1 - (χ ^ 2) ↑↑p * ↑↑p ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
|
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
hsum₂ : Summable fun x_1 => (-(1 - (χ ^ 2) ↑↑x_1 * ↑↑x_1 ^ (-(1 + ↑x + 2 * I * ↑y))).log).re
⊢ 0 ≤
∑' (b : Primes),
(3 * (-(1 - χ₀ ↑↑b * ↑↑b ^ (-(1 + ↑x))).log).re + 4 * (-(1 - χ ↑↑b * ↑↑b ^ (-(1 + ↑x + I * ↑y))).log).re +
(-(1 - (χ ^ 2) ↑↑b * ↑↑b ^ (-(1 + ↑x + 2 * I * ↑y))).log).re)
|
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