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/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)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
convert tsum_nonneg fun p : Nat.Primes ↦ re_log_comb_nonneg_dirichlet χ p.prop.two_le h₀
|
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)
|
case h.e'_4.h.e'_5.h.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6.h.e'_5
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
x✝ : Primes
⊢ (χ ^ 2) ↑↑x✝ = χ ↑↑x✝ ^ 2
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
rw [sq, sq, MulChar.mul_apply]
|
case h.e'_4.h.e'_5.h.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6.h.e'_5
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
x✝ : Primes
⊢ (χ ^ 2) ↑↑x✝ = χ ↑↑x✝ ^ 2
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_dirichlet_product_ge_one
|
[147, 1]
|
[174, 33]
|
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
⊢ 1 + ↑x = ↑(1 + ↑x).re
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_zeta_product_ge_one
|
[176, 1]
|
[182, 88]
|
have ⟨h₀, h₁, h₂⟩ := one_lt_re_of_pos y hx
|
x : ℝ
hx : 0 < x
y : ℝ
⊢ ‖ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑y) ^ 4 * ζ (1 + ↑x + 2 * I * ↑y)‖ ≥ 1
|
x : ℝ
hx : 0 < x
y : ℝ
h₀ : 1 < (1 + ↑x).re
h₁ : 1 < (1 + ↑x + I * ↑y).re
h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re
⊢ ‖ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑y) ^ 4 * ζ (1 + ↑x + 2 * I * ↑y)‖ ≥ 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
norm_zeta_product_ge_one
|
[176, 1]
|
[182, 88]
|
simpa only [one_pow, norm_mul, norm_pow, DirichletCharacter.LSeries_modOne_eq,
LSeries_one_eq_riemannZeta, h₀, h₁, h₂] using norm_dirichlet_product_ge_one χ₁ hx y
|
x : ℝ
hx : 0 < x
y : ℝ
h₀ : 1 < (1 + ↑x).re
h₁ : 1 < (1 + ↑x + I * ↑y).re
h₂ : 1 < (1 + ↑x + 2 * I * ↑y).re
⊢ ‖ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑y) ^ 4 * ζ (1 + ↑x + 2 * I * ↑y)‖ ≥ 1
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
rw [isBigO_iff', isBigO_iff']
|
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
⊢ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x
|
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;>
{ refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_
rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]
have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx
rw [le_div_iff hx', mul_comm] }
|
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
hx' : ‖f x‖ > 0
⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
Asymptotics.isBigO_mul_iff_isBigO_div
|
[251, 1]
|
[259, 36]
|
rw [le_div_iff hx', mul_comm]
|
case refine_2
α : Type u_1
F : Type u_2
inst✝ : NormedField F
l : Filter α
f g h : α → F
hf : ∀ᶠ (x : α) in l, f x ≠ 0
x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
c : ℝ
hc : c > 0
H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
x : α
hx : f x ≠ 0
hx' : ‖f x‖ > 0
⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
DifferentiableAt.isBigO_of_eq_zero
|
[269, 1]
|
[273, 73]
|
rw [← zero_add z] at hf
|
f : ℂ → ℂ
z : ℂ
hf : DifferentiableAt ℂ f z
hz : f z = 0
⊢ (fun w => f (w + z)) =O[𝓝 0] id
|
f : ℂ → ℂ
z : ℂ
hf : DifferentiableAt ℂ f (0 + z)
hz : f z = 0
⊢ (fun w => f (w + z)) =O[𝓝 0] id
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
DifferentiableAt.isBigO_of_eq_zero
|
[269, 1]
|
[273, 73]
|
simpa only [zero_add, hz, sub_zero]
using (hf.hasDerivAt.comp_add_const 0 z).differentiableAt.isBigO_sub
|
f : ℂ → ℂ
z : ℂ
hf : DifferentiableAt ℂ f (0 + z)
hz : f z = 0
⊢ (fun w => f (w + z)) =O[𝓝 0] id
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
rw [isBigO_iff']
|
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ (fun w => f (w + z)) =O[𝓝 0] fun x => 1
|
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
simp_rw [Metric.continuousAt_iff', dist_eq_norm_sub, zero_add] at hf
|
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt (fun w => f (w + z)) 0
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
f : ℂ → ℂ
z : ℂ
hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
specialize hf 1 zero_lt_one
|
f : ℂ → ℂ
z : ℂ
hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
refine ⟨‖f z‖ + 1, by positivity, ?_⟩
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
refine Eventually.mp hf <| eventually_of_forall fun w hw ↦ le_of_lt ?_
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
w : ℂ
hw : ‖f (w + z) - f z‖ < 1
⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
calc ‖f (w + z)‖
_ ≤ ‖f z‖ + ‖f (w + z) - f z‖ := norm_le_insert' ..
_ < ‖f z‖ + 1 := add_lt_add_left hw _
_ = _ := by simp only [norm_one, mul_one]
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
w : ℂ
hw : ‖f (w + z) - f z‖ < 1
⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
convert (Homeomorph.comp_continuousAt_iff' (Homeomorph.addLeft (-z)) _ z).mp ?_
|
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ ContinuousAt (fun w => f (w + z)) 0
|
case h.e'_1
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ 0 = (Homeomorph.addLeft (-z)) z
case convert_4
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
simp
|
case h.e'_1
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ 0 = (Homeomorph.addLeft (-z)) z
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
simp [Function.comp_def, hf]
|
case convert_4
f : ℂ → ℂ
z : ℂ
hf : ContinuousAt f z
⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
positivity
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
⊢ ‖f z‖ + 1 > 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ContinuousAt.isBigO
|
[275, 1]
|
[289, 46]
|
simp only [norm_one, mul_one]
|
f : ℂ → ℂ
z : ℂ
hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1
w : ℂ
hw : ‖f (w + z) - f z‖ < 1
⊢ ‖f z‖ + 1 = (‖f z‖ + 1) * ‖1‖
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
exact (isBigO_comp_ofReal_nhds_ne this).mono <| nhds_right'_le_nhds_ne 0
|
this : (fun w => ζ (1 + w)) =O[𝓝[≠] 0] fun x => 1 / x
⊢ (fun x => ζ (1 + ↑x)) =O[𝓝[>] 0] fun x => 1 / ↑x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
exact ((isBigO_mul_iff_isBigO_div eventually_mem_nhdsWithin).mp <|
Tendsto.isBigO_one ℂ H).trans <| isBigO_refl ..
|
H : Tendsto (fun w => w * ζ (1 + w)) (𝓝[≠] 0) (𝓝 1)
⊢ (fun w => ζ (1 + w)) =O[𝓝[≠] 0] fun x => 1 / x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
convert Tendsto.comp (f := fun w ↦ 1 + w) riemannZeta_residue_one ?_ using 1
|
⊢ Tendsto (fun w => w * ζ (1 + w)) (𝓝[≠] 0) (𝓝 1)
|
case h.e'_3
⊢ (fun w => w * ζ (1 + w)) = (fun s => (s - 1) * ζ s) ∘ fun w => 1 + w
case convert_2
⊢ Tendsto (fun w => 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
ext w
|
case h.e'_3
⊢ (fun w => w * ζ (1 + w)) = (fun s => (s - 1) * ζ s) ∘ fun w => 1 + w
|
case h.e'_3.h
w : ℂ
⊢ w * ζ (1 + w) = ((fun s => (s - 1) * ζ s) ∘ fun w => 1 + w) w
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
simp only [Function.comp_apply, add_sub_cancel_left]
|
case h.e'_3.h
w : ℂ
⊢ w * ζ (1 + w) = ((fun s => (s - 1) * ζ s) ∘ fun w => 1 + w) w
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
refine tendsto_iff_comap.mpr <| map_le_iff_le_comap.mp <| Eq.le ?_
|
case convert_2
⊢ Tendsto (fun w => 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)
|
case convert_2
⊢ map (fun w => 1 + w) (𝓝[≠] 0) = 𝓝[≠] 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_one_horizontal
|
[307, 1]
|
[318, 75]
|
convert map_punctured_nhds_eq (Homeomorph.addLeft (1 : ℂ)) 0 using 2 <;> simp
|
case convert_2
⊢ map (fun w => 1 + w) (𝓝[≠] 0) = 𝓝[≠] 1
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_of_ne_one_horizontal
|
[320, 1]
|
[326, 7]
|
refine Asymptotics.IsBigO.mono ?_ nhdsWithin_le_nhds
|
y : ℝ
hy : y ≠ 0
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝[>] 0] fun x => 1
|
y : ℝ
hy : y ≠ 0
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝 0] fun x => 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_of_ne_one_horizontal
|
[320, 1]
|
[326, 7]
|
have hy' : 1 + I * y ≠ 1 := by simp [hy]
|
y : ℝ
hy : y ≠ 0
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝 0] fun x => 1
|
y : ℝ
hy : y ≠ 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝 0] fun x => 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_of_ne_one_horizontal
|
[320, 1]
|
[326, 7]
|
convert isBigO_comp_ofReal
(differentiableAt_riemannZeta hy').continuousAt.isBigO using 3 with x
|
y : ℝ
hy : y ≠ 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝 0] fun x => 1
|
case h.e'_7.h.h.e'_1
y : ℝ
hy : y ≠ 0
hy' : 1 + I * ↑y ≠ 1
x : ℝ
⊢ 1 + ↑x + I * ↑y = ↑x + (1 + I * ↑y)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_of_ne_one_horizontal
|
[320, 1]
|
[326, 7]
|
ring
|
case h.e'_7.h.h.e'_1
y : ℝ
hy : y ≠ 0
hy' : 1 + I * ↑y ≠ 1
x : ℝ
⊢ 1 + ↑x + I * ↑y = ↑x + (1 + I * ↑y)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_of_ne_one_horizontal
|
[320, 1]
|
[326, 7]
|
simp [hy]
|
y : ℝ
hy : y ≠ 0
⊢ 1 + I * ↑y ≠ 1
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_root_horizontal
|
[328, 1]
|
[333, 23]
|
have hy' : 1 + I * y ≠ 1 := by simp [hy]
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝[>] 0] fun x => ↑x
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝[>] 0] fun x => ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_root_horizontal
|
[328, 1]
|
[333, 23]
|
conv => enter [2, x]; rw [add_comm 1, add_assoc]
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (1 + ↑x + I * ↑y)) =O[𝓝[>] 0] fun x => ↑x
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (↑x + (1 + I * ↑y))) =O[𝓝[>] 0] fun x => ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_root_horizontal
|
[328, 1]
|
[333, 23]
|
exact (isBigO_comp_ofReal <| (differentiableAt_riemannZeta hy').isBigO_of_eq_zero h).mono
nhdsWithin_le_nhds
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
hy' : 1 + I * ↑y ≠ 1
⊢ (fun x => ζ (↑x + (1 + I * ↑y))) =O[𝓝[>] 0] fun x => ↑x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_isBigO_near_root_horizontal
|
[328, 1]
|
[333, 23]
|
simp [hy]
|
y : ℝ
hy : y ≠ 0
h : ζ (1 + I * ↑y) = 0
⊢ 1 + I * ↑y ≠ 1
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
refine hz'.eq_or_lt.elim (fun h Hz ↦ ?_) riemannZeta_ne_zero_of_one_lt_re
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
⊢ ζ z ≠ 0
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
have hz₀ : z.im ≠ 0 := by
rw [← re_add_im z, ← h, ofReal_one] at hz
simpa only [ne_eq, add_right_eq_self, mul_eq_zero, ofReal_eq_zero, I_ne_zero, or_false]
using hz
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
have hzeq : z = 1 + I * z.im := by
rw [mul_comm I, ← re_add_im z, ← h]
push_cast
simp only [add_im, one_im, mul_im, ofReal_re, I_im, mul_one, ofReal_im, I_re, mul_zero,
add_zero, zero_add]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
have H₀ : (fun _ : ℝ ↦ (1 : ℝ)) =O[𝓝[>] 0]
(fun x ↦ ζ (1 + x) ^ 3 * ζ (1 + x + I * z.im) ^ 4 * ζ (1 + x + 2 * I * z.im)) :=
IsBigO.of_bound' <| eventually_nhdsWithin_of_forall
fun _ hx ↦ (norm_one (α := ℝ)).symm ▸ (norm_zeta_product_ge_one hx z.im).le
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
have H := (riemannZeta_isBigO_near_one_horizontal.pow 3).mul
((riemannZeta_isBigO_near_root_horizontal hz₀ (hzeq ▸ Hz)).pow 4)|>.mul <|
riemannZeta_isBigO_of_ne_one_horizontal <| mul_ne_zero two_ne_zero hz₀
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
conv at H => enter [3, x]; rw [help]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
conv at H =>
enter [2, x]; rw [show 1 + x + I * ↑(2 * z.im) = 1 + x + 2 * I * z.im by simp; ring]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
replace H := (H₀.trans H).norm_right
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => ‖↑x‖
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
simp only [norm_eq_abs, abs_ofReal] at H
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => ‖↑x‖
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
refine isLittleO_irrefl ?_ <| H.of_abs_right.trans_isLittleO <|
isLittleO_id_nhdsWithin (Set.Ioi 0)
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ False
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ ∃ᶠ (x : ℝ) in 𝓝[>] 0, 1 ≠ 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
simp only [ne_eq, one_ne_zero, not_false_eq_true, frequently_true_iff_neBot]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ ∃ᶠ (x : ℝ) in 𝓝[>] 0, 1 ≠ 0
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ (𝓝[>] 0).NeBot
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
exact mem_closure_iff_nhdsWithin_neBot.mp <| closure_Ioi (0 : ℝ) ▸ Set.left_mem_Ici
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
H : (fun x => 1) =O[𝓝[>] 0] fun x => |x|
⊢ (𝓝[>] 0).NeBot
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
rw [← re_add_im z, ← h, ofReal_one] at hz
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
⊢ z.im ≠ 0
|
z : ℂ
hz : 1 + ↑z.im * I ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
⊢ z.im ≠ 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
simpa only [ne_eq, add_right_eq_self, mul_eq_zero, ofReal_eq_zero, I_ne_zero, or_false]
using hz
|
z : ℂ
hz : 1 + ↑z.im * I ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
⊢ z.im ≠ 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
rw [mul_comm I, ← re_add_im z, ← h]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ z = 1 + I * ↑z.im
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ ↑1 + ↑z.im * I = 1 + ↑(↑1 + ↑z.im * I).im * I
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
push_cast
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ ↑1 + ↑z.im * I = 1 + ↑(↑1 + ↑z.im * I).im * I
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ 1 + ↑z.im * I = 1 + ↑(1 + ↑z.im * I).im * I
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
simp only [add_im, one_im, mul_im, ofReal_re, I_im, mul_one, ofReal_im, I_re, mul_zero,
add_zero, zero_add]
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
⊢ 1 + ↑z.im * I = 1 + ↑(1 + ↑z.im * I).im * I
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
rcases eq_or_ne x 0 with rfl | h
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
x : ℝ
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
|
case inl
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
⊢ (1 / ↑0) ^ 3 * ↑0 ^ 4 * 1 = ↑0
case inr
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h✝ : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
x : ℝ
h : x ≠ 0
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
rw [ofReal_zero, zero_pow (by norm_num), mul_zero, mul_one]
|
case inl
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
⊢ (1 / ↑0) ^ 3 * ↑0 ^ 4 * 1 = ↑0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
norm_num
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
⊢ 4 ≠ 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
field_simp [h]
|
case inr
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h✝ : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
x : ℝ
h : x ≠ 0
⊢ (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
|
case inr
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h✝ : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
x : ℝ
h : x ≠ 0
⊢ ↑x ^ 4 = ↑x * ↑x ^ 3
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
ring
|
case inr
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h✝ : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H :
(fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x =>
(1 / ↑x) ^ 3 * ↑x ^ 4 * 1
x : ℝ
h : x ≠ 0
⊢ ↑x ^ 4 = ↑x * ↑x ^ 3
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
simp
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
x : ℝ
⊢ 1 + ↑x + I * ↑(2 * z.im) = 1 + ↑x + 2 * I * ↑z.im
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
x : ℝ
⊢ I * (2 * ↑z.im) = 2 * I * ↑z.im
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
riemannZeta_ne_zero_of_one_le_re
|
[335, 1]
|
[370, 86]
|
ring
|
z : ℂ
hz : z ≠ 1
hz' : 1 ≤ z.re
h : 1 = z.re
Hz : ζ z = 0
hz₀ : z.im ≠ 0
hzeq : z = 1 + I * ↑z.im
H₀ : (fun x => 1) =O[𝓝[>] 0] fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + 2 * I * ↑z.im)
H : (fun x => ζ (1 + ↑x) ^ 3 * ζ (1 + ↑x + I * ↑z.im) ^ 4 * ζ (1 + ↑x + I * ↑(2 * z.im))) =O[𝓝[>] 0] fun x => ↑x
help : ∀ (x : ℝ), (1 / ↑x) ^ 3 * ↑x ^ 4 * 1 = ↑x
x : ℝ
⊢ I * (2 * ↑z.im) = 2 * I * ↑z.im
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_apply_of_ne_one
|
[386, 1]
|
[387, 16]
|
simp [ζ₁, hz]
|
z : ℂ
hz : z ≠ 1
⊢ ζ₁ z = ζ z * (z - 1)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
rw [← differentiableOn_univ,
← differentiableOn_compl_singleton_and_continuousAt_iff (c := 1) Filter.univ_mem, ζ₁]
|
⊢ Differentiable ℂ ζ₁
|
⊢ DifferentiableOn ℂ (Function.update (fun z => ζ z * (z - 1)) 1 1) (Set.univ \ {1}) ∧
ContinuousAt (Function.update (fun z => ζ z * (z - 1)) 1 1) 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
refine ⟨DifferentiableOn.congr (f := fun z ↦ ζ z * (z - 1))
(fun _ hz ↦ DifferentiableAt.differentiableWithinAt ?_) fun _ hz ↦ ?_,
continuousWithinAt_compl_self.mp ?_⟩
|
⊢ DifferentiableOn ℂ (Function.update (fun z => ζ z * (z - 1)) 1 1) (Set.univ \ {1}) ∧
ContinuousAt (Function.update (fun z => ζ z * (z - 1)) 1 1) 1
|
case refine_1
x✝ : ℂ
hz : x✝ ∈ Set.univ \ {1}
⊢ DifferentiableAt ℂ (fun z => ζ z * (z - 1)) x✝
case refine_2
x✝ : ℂ
hz : x✝ ∈ Set.univ \ {1}
⊢ Function.update (fun z => ζ z * (z - 1)) 1 1 x✝ = (fun z => ζ z * (z - 1)) x✝
case refine_3
⊢ ContinuousWithinAt (Function.update (fun z => ζ z * (z - 1)) 1 1) {1}ᶜ 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
simp only [Set.mem_diff, Set.mem_univ, Set.mem_singleton_iff, true_and] at hz
|
case refine_1
x✝ : ℂ
hz : x✝ ∈ Set.univ \ {1}
⊢ DifferentiableAt ℂ (fun z => ζ z * (z - 1)) x✝
|
case refine_1
x✝ : ℂ
hz : ¬x✝ = 1
⊢ DifferentiableAt ℂ (fun z => ζ z * (z - 1)) x✝
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
exact (differentiableAt_riemannZeta hz).mul <| (differentiableAt_id').sub <|
differentiableAt_const 1
|
case refine_1
x✝ : ℂ
hz : ¬x✝ = 1
⊢ DifferentiableAt ℂ (fun z => ζ z * (z - 1)) x✝
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
simp only [Set.mem_diff, Set.mem_univ, Set.mem_singleton_iff, true_and] at hz
|
case refine_2
x✝ : ℂ
hz : x✝ ∈ Set.univ \ {1}
⊢ Function.update (fun z => ζ z * (z - 1)) 1 1 x✝ = (fun z => ζ z * (z - 1)) x✝
|
case refine_2
x✝ : ℂ
hz : ¬x✝ = 1
⊢ Function.update (fun z => ζ z * (z - 1)) 1 1 x✝ = (fun z => ζ z * (z - 1)) x✝
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
simp only [ne_eq, hz, not_false_eq_true, Function.update_noteq]
|
case refine_2
x✝ : ℂ
hz : ¬x✝ = 1
⊢ Function.update (fun z => ζ z * (z - 1)) 1 1 x✝ = (fun z => ζ z * (z - 1)) x✝
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
conv in (_ * _) => rw [mul_comm]
|
case refine_3
⊢ ContinuousWithinAt (Function.update (fun z => ζ z * (z - 1)) 1 1) {1}ᶜ 1
|
case refine_3
⊢ ContinuousWithinAt (Function.update (fun z => (z - 1) * ζ z) 1 1) {1}ᶜ 1
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
simp only [continuousWithinAt_compl_self, continuousAt_update_same]
|
case refine_3
⊢ ContinuousWithinAt (Function.update (fun z => (z - 1) * ζ z) 1 1) {1}ᶜ 1
|
case refine_3
⊢ Filter.Tendsto (fun z => (z - 1) * ζ z) (nhdsWithin 1 {1}ᶜ) (nhds 1)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
ζ₁_differentiable
|
[389, 1]
|
[402, 34]
|
exact riemannZeta_residue_one
|
case refine_3
⊢ Filter.Tendsto (fun z => (z - 1) * ζ z) (nhdsWithin 1 {1}ᶜ) (nhds 1)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
deriv_ζ₁_apply_of_ne_one
|
[404, 1]
|
[411, 68]
|
have H : deriv ζ₁ z = deriv (fun w ↦ ζ w * (w - 1)) z := by
refine Filter.EventuallyEq.deriv_eq <| Filter.eventuallyEq_iff_exists_mem.mpr ?_
refine ⟨{w | w ≠ 1}, IsOpen.mem_nhds isOpen_ne hz, fun w hw ↦ ?_⟩
simp only [ζ₁, ne_eq, Set.mem_setOf.mp hw, not_false_eq_true, Function.update_noteq]
|
z : ℂ
hz : z ≠ 1
⊢ deriv ζ₁ z = deriv ζ z * (z - 1) + ζ z
|
z : ℂ
hz : z ≠ 1
H : deriv ζ₁ z = deriv (fun w => ζ w * (w - 1)) z
⊢ deriv ζ₁ z = deriv ζ z * (z - 1) + ζ z
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
deriv_ζ₁_apply_of_ne_one
|
[404, 1]
|
[411, 68]
|
rw [H, deriv_mul (differentiableAt_riemannZeta hz) <| differentiableAt_id'.sub <|
differentiableAt_const 1, deriv_sub_const, deriv_id'', mul_one]
|
z : ℂ
hz : z ≠ 1
H : deriv ζ₁ z = deriv (fun w => ζ w * (w - 1)) z
⊢ deriv ζ₁ z = deriv ζ z * (z - 1) + ζ z
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
deriv_ζ₁_apply_of_ne_one
|
[404, 1]
|
[411, 68]
|
refine Filter.EventuallyEq.deriv_eq <| Filter.eventuallyEq_iff_exists_mem.mpr ?_
|
z : ℂ
hz : z ≠ 1
⊢ deriv ζ₁ z = deriv (fun w => ζ w * (w - 1)) z
|
z : ℂ
hz : z ≠ 1
⊢ ∃ s ∈ nhds z, Set.EqOn ζ₁ (fun w => ζ w * (w - 1)) s
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
deriv_ζ₁_apply_of_ne_one
|
[404, 1]
|
[411, 68]
|
refine ⟨{w | w ≠ 1}, IsOpen.mem_nhds isOpen_ne hz, fun w hw ↦ ?_⟩
|
z : ℂ
hz : z ≠ 1
⊢ ∃ s ∈ nhds z, Set.EqOn ζ₁ (fun w => ζ w * (w - 1)) s
|
z : ℂ
hz : z ≠ 1
w : ℂ
hw : w ∈ {w | w ≠ 1}
⊢ ζ₁ w = (fun w => ζ w * (w - 1)) w
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
deriv_ζ₁_apply_of_ne_one
|
[404, 1]
|
[411, 68]
|
simp only [ζ₁, ne_eq, Set.mem_setOf.mp hw, not_false_eq_true, Function.update_noteq]
|
z : ℂ
hz : z ≠ 1
w : ℂ
hw : w ∈ {w | w ≠ 1}
⊢ ζ₁ w = (fun w => ζ w * (w - 1)) w
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
neg_logDeriv_ζ₁_eq
|
[413, 1]
|
[417, 7]
|
rw [ζ₁_apply_of_ne_one hz₁, deriv_ζ₁_apply_of_ne_one hz₁]
|
z : ℂ
hz₁ : z ≠ 1
hz₂ : ζ z ≠ 0
⊢ -deriv ζ₁ z / ζ₁ z = -deriv ζ z / ζ z - 1 / (z - 1)
|
z : ℂ
hz₁ : z ≠ 1
hz₂ : ζ z ≠ 0
⊢ -(deriv ζ z * (z - 1) + ζ z) / (ζ z * (z - 1)) = -deriv ζ z / ζ z - 1 / (z - 1)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
neg_logDeriv_ζ₁_eq
|
[413, 1]
|
[417, 7]
|
field_simp [sub_ne_zero.mpr hz₁]
|
z : ℂ
hz₁ : z ≠ 1
hz₂ : ζ z ≠ 0
⊢ -(deriv ζ z * (z - 1) + ζ z) / (ζ z * (z - 1)) = -deriv ζ z / ζ z - 1 / (z - 1)
|
z : ℂ
hz₁ : z ≠ 1
hz₂ : ζ z ≠ 0
⊢ -ζ z + -(deriv ζ z * (z - 1)) = -(deriv ζ z * (z - 1)) - ζ z
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
neg_logDeriv_ζ₁_eq
|
[413, 1]
|
[417, 7]
|
ring
|
z : ℂ
hz₁ : z ≠ 1
hz₂ : ζ z ≠ 0
⊢ -ζ z + -(deriv ζ z * (z - 1)) = -(deriv ζ z * (z - 1)) - ζ z
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
simp_rw [neg_div]
|
⊢ ContinuousOn (fun z => -deriv ζ₁ z / ζ₁ z) {z | z = 1 ∨ ζ z ≠ 0}
|
⊢ ContinuousOn (fun z => -(deriv ζ₁ z / ζ₁ z)) {z | z = 1 ∨ ζ z ≠ 0}
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
refine ((ζ₁_differentiable.contDiff.continuous_deriv le_rfl).continuousOn.div
ζ₁_differentiable.continuous.continuousOn fun w hw ↦ ?_).neg
|
⊢ ContinuousOn (fun z => -(deriv ζ₁ z / ζ₁ z)) {z | z = 1 ∨ ζ z ≠ 0}
|
w : ℂ
hw : w ∈ {z | z = 1 ∨ ζ z ≠ 0}
⊢ ζ₁ w ≠ 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
rcases eq_or_ne w 1 with rfl | hw'
|
w : ℂ
hw : w ∈ {z | z = 1 ∨ ζ z ≠ 0}
⊢ ζ₁ w ≠ 0
|
case inl
hw : 1 ∈ {z | z = 1 ∨ ζ z ≠ 0}
⊢ ζ₁ 1 ≠ 0
case inr
w : ℂ
hw : w ∈ {z | z = 1 ∨ ζ z ≠ 0}
hw' : w ≠ 1
⊢ ζ₁ w ≠ 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
simp only [ζ₁, Function.update_same, ne_eq, one_ne_zero, not_false_eq_true]
|
case inl
hw : 1 ∈ {z | z = 1 ∨ ζ z ≠ 0}
⊢ ζ₁ 1 ≠ 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
simp only [ne_eq, Set.mem_setOf_eq, hw', false_or] at hw
|
case inr
w : ℂ
hw : w ∈ {z | z = 1 ∨ ζ z ≠ 0}
hw' : w ≠ 1
⊢ ζ₁ w ≠ 0
|
case inr
w : ℂ
hw' : w ≠ 1
hw : ¬ζ w = 0
⊢ ζ₁ w ≠ 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
simp only [ζ₁, ne_eq, hw', not_false_eq_true, Function.update_noteq, _root_.mul_eq_zero, hw,
false_or]
|
case inr
w : ℂ
hw' : w ≠ 1
hw : ¬ζ w = 0
⊢ ζ₁ w ≠ 0
|
case inr
w : ℂ
hw' : w ≠ 1
hw : ¬ζ w = 0
⊢ ¬w - 1 = 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
continuousOn_neg_logDeriv_ζ₁
|
[419, 1]
|
[429, 30]
|
exact sub_ne_zero.mpr hw'
|
case inr
w : ℂ
hw' : w ≠ 1
hw : ¬ζ w = 0
⊢ ¬w - 1 = 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
have hnv := riemannZeta_ne_zero_of_one_le_re
|
WIT : WienerIkeharaTheorem
⊢ Tendsto (fun N => (Finset.range N).sum ⇑Λ / ↑N) atTop (nhds 1)
|
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
⊢ Tendsto (fun N => (Finset.range N).sum ⇑Λ / ↑N) atTop (nhds 1)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
refine WIT (F := fun z ↦ -deriv ζ₁ z / ζ₁ z) (fun _ ↦ vonMangoldt_nonneg) (fun s hs ↦ ?_) ?_
|
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
⊢ Tendsto (fun N => (Finset.range N).sum ⇑Λ / ↑N) atTop (nhds 1)
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
⊢ (fun z => -deriv ζ₁ z / ζ₁ z) s = (fun s => L (fun n => ↑(Λ n)) s - ↑1 / (s - 1)) s
case refine_2
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
⊢ ContinuousOn (fun z => -deriv ζ₁ z / ζ₁ z) {s | 1 ≤ s.re}
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
have hs₁ : s ≠ 1 := by
rintro rfl
simp at hs
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
⊢ (fun z => -deriv ζ₁ z / ζ₁ z) s = (fun s => L (fun n => ↑(Λ n)) s - ↑1 / (s - 1)) s
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
hs₁ : s ≠ 1
⊢ (fun z => -deriv ζ₁ z / ζ₁ z) s = (fun s => L (fun n => ↑(Λ n)) s - ↑1 / (s - 1)) s
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
simp only [ne_eq, hs₁, not_false_eq_true, LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs,
ofReal_one]
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
hs₁ : s ≠ 1
⊢ (fun z => -deriv ζ₁ z / ζ₁ z) s = (fun s => L (fun n => ↑(Λ n)) s - ↑1 / (s - 1)) s
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
hs₁ : s ≠ 1
⊢ -deriv ζ₁ s / ζ₁ s = -deriv ζ s / ζ s - 1 / (s - 1)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
exact neg_logDeriv_ζ₁_eq hs₁ <| hnv hs₁ (Set.mem_setOf.mp hs).le
|
case refine_1
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
hs₁ : s ≠ 1
⊢ -deriv ζ₁ s / ζ₁ s = -deriv ζ s / ζ s - 1 / (s - 1)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
rintro rfl
|
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
s : ℂ
hs : s ∈ {s | 1 < s.re}
⊢ s ≠ 1
|
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
hs : 1 ∈ {s | 1 < s.re}
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/PNT.lean
|
PNT_vonMangoldt
|
[438, 1]
|
[451, 10]
|
simp at hs
|
WIT : WienerIkeharaTheorem
hnv : ∀ ⦃z : ℂ⦄, z ≠ 1 → 1 ≤ z.re → ζ z ≠ 0
hs : 1 ∈ {s | 1 < s.re}
⊢ False
|
no goals
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.