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stringclasses 147
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stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
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stringlengths 1
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stringlengths 3
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2.09M
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|---|---|---|---|---|---|---|---|---|
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 |