mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case inr.refine_2
k✝ : ℕ
hk✝ : k✝ ≠ 0
n : ℤ
hn : n ≠ 0
k : ℕ
hk : 1 ≤ k
h'k : bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
⊢ bernoulliFourierCoeff (k + 1) n = -↑(k + 1)! / (2 * ↑π * I * ↑n) ^ (k + 1)

Tactic:
·

Goal state after:
case inr.refine_2
k✝ : ℕ
hk✝ : k✝ ≠ 0
n : ℤ
hn : n ≠ 0
k : ℕ
hk : 1 ≤ k
h'k : bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
⊢ bernoulliFourierCoeff (k + 1) n = -↑(k + 1)! / (2 * ↑π * I * ↑n) ^ (k + 1)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_2
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k))

pp: case e'_3
k : ℕ
hk : k ≠ 0
⊢ (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)! =
    ↑((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!)

Tactic:
·

Goal state after:
case e'_2
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
hk : k ≠ 0
⊢ 1 < 2 * k

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
⊢ completedLFunction Φ s = completedLFunction₀ Φ s - ↑N ^ (-s) * Φ 0 / s - (↑N ^ (-s) * ∑ j, Φ j) / (1 - s)

Tactic:
simp only [completedLFunction, completedHurwitzZetaEven_eq, toAddCircle_eq_zero, div_eq_mul_inv,
    ite_mul, one_mul, zero_mul, mul_sub, mul_ite, mul_zero, sum_sub_distrib, Fintype.sum_ite_eq',
    ← sum_mul, completedLFunction₀, mul_assoc]
  abel

Goal state after:
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_3
k : ℕ
hk : k ≠ 0
⊢ (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)! =
    ↑((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!)

Tactic:
·

Goal state after:
case e'_3
k : ℕ
hk : k ≠ 0
⊢ (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)! =
    ↑((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x) * (Φ 0 / x)) s

pp: case refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x) * ((∑ j, Φ j) / (1 - x))) s

Tactic:
·

Goal state after:
case refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x) * (Φ 0 / x)) s
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_1.refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ s ≠ 0 → DifferentiableAt ℂ (HDiv.hDiv (Φ 0)) s

pp: case refine_1.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ Φ 0 = 0 → DifferentiableAt ℂ (HDiv.hDiv (Φ 0)) s

Tactic:
·

Goal state after:
case refine_1.refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ s ≠ 0 → DifferentiableAt ℂ (HDiv.hDiv (Φ 0)) s
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_1.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ Φ 0 = 0 → DifferentiableAt ℂ (HDiv.hDiv (Φ 0)) s

Tactic:
·

Goal state after:
case refine_1.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ Φ 0 = 0 → DifferentiableAt ℂ (HDiv.hDiv (Φ 0)) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case inl
⊢ riemannZeta (2 * ↑0) = (-1) ^ (0 + 1) * 2 ^ (2 * ↑0 - 1) * ↑π ^ (2 * 0) * ↑(bernoulli (2 * 0)) / ↑(2 * 0)!

pp: case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (2 * ↑k) = (-1) ^ (k + 1) * 2 ^ (2 * ↑k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!

Tactic:
·

Goal state after:
case inl
⊢ riemannZeta (2 * ↑0) = (-1) ^ (0 + 1) * 2 ^ (2 * ↑0 - 1) * ↑π ^ (2 * 0) * ↑(bernoulli (2 * 0)) / ↑(2 * 0)!
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x) * ((∑ j, Φ j) / (1 - x))) s

Tactic:
·

Goal state after:
case refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun x ↦ ↑N ^ (-x) * ((∑ j, Φ j) / (1 - x))) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (2 * ↑k) = (-1) ^ (k + 1) * 2 ^ (2 * ↑k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!

Tactic:
·

Goal state after:
case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (2 * ↑k) = (-1) ^ (k + 1) * 2 ^ (2 * ↑k - 1) * ↑π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_2.refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ s ≠ 1 → DifferentiableAt ℂ (fun x ↦ (∑ j, Φ j) / (1 - x)) s

pp: case refine_2.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ ∑ j, Φ j = 0 → DifferentiableAt ℂ (fun x ↦ (∑ j, Φ j) / (1 - x)) s

Tactic:
·

Goal state after:
case refine_2.refine_1
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ s ≠ 1 → DifferentiableAt ℂ (fun x ↦ (∑ j, Φ j) / (1 - x)) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_2
⊢ riemannZeta 2 = ↑(∑' (b : ℕ), 1 / ↑b ^ 2)

pp: case e'_3
⊢ ↑π ^ 2 / 6 = ↑(π ^ 2 / 6)

Tactic:
·

Goal state after:
case e'_2
⊢ riemannZeta 2 = ↑(∑' (b : ℕ), 1 / ↑b ^ 2)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_3
⊢ ↑π ^ 2 / 6 = ↑(π ^ 2 / 6)

Tactic:
·

Goal state after:
case e'_3
⊢ ↑π ^ 2 / 6 = ↑(π ^ 2 / 6)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_2
⊢ riemannZeta 4 = ↑(∑' (b : ℕ), 1 / ↑b ^ 4)

pp: case e'_3
⊢ ↑π ^ 4 / 90 = ↑(π ^ 4 / 90)

Tactic:
·

Goal state after:
case e'_2
⊢ riemannZeta 4 = ↑(∑' (b : ℕ), 1 / ↑b ^ 4)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case e'_3
⊢ ↑π ^ 4 / 90 = ↑(π ^ 4 / 90)

Tactic:
·

Goal state after:
case e'_3
⊢ ↑π ^ 4 / 90 = ↑(π ^ 4 / 90)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case refine_2.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ ∑ j, Φ j = 0 → DifferentiableAt ℂ (fun x ↦ (∑ j, Φ j) / (1 - x)) s

Tactic:
·

Goal state after:
case refine_2.refine_2
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs₀ : s ≠ 0 ∨ Φ 0 = 0
hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0
⊢ ∑ j, Φ j = 0 → DifferentiableAt ℂ (fun x ↦ (∑ j, Φ j) / (1 - x)) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case inl
⊢ riemannZeta (-↑0) = -↑(bernoulli' (0 + 1)) / (↑0 + 1)

pp: case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (-↑k) = -↑(bernoulli' (k + 1)) / (↑k + 1)

Tactic:
·

Goal state after:
case inl
⊢ riemannZeta (-↑0) = -↑(bernoulli' (0 + 1)) / (↑0 + 1)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (-↑k) = -↑(bernoulli' (k + 1)) / (↑k + 1)

Tactic:
·

Goal state after:
case inr
k : ℕ
hk : k ≠ 0
⊢ riemannZeta (-↑k) = -↑(bernoulli' (k + 1)) / (↑k + 1)
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case inl
k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
hx : x ∈ Ico 0 1
⊢ HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))

pp: case inr
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑1) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1))

Tactic:
·

Goal state after:
case inl
k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
hx : x ∈ Ico 0 1
⊢ HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case inr
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑1) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1))

Tactic:
·

Goal state after:
case inr
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑1) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1))
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑1) = fun n ↦ 1 / ↑n ^ k * (fourier n) ↑0

pp: case e'_6
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 0)

Tactic:
·

Goal state after:
case e'_5
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑1) = fun n ↦ 1 / ↑n ^ k * (fourier n) ↑0
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case e_a.e_f.inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0 ∨ Φ 0 = 0
i : ZMod N
hi : i ≠ 0
⊢ Φ i * hurwitzZetaEven (toAddCircle i) s = Φ i * (completedHurwitzZetaEven (toAddCircle i) s / s.Gammaℝ)

pp: case e_a.e_f.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0 ∨ Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)

Tactic:
·

Goal state after:
case e_a.e_f.inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0 ∨ Φ 0 = 0
i : ZMod N
hi : i ≠ 0
⊢ Φ i * hurwitzZetaEven (toAddCircle i) s = Φ i * (completedHurwitzZetaEven (toAddCircle i) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_6
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 0)

Tactic:
·

Goal state after:
case e'_6
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 0)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case e_a.e_f.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0 ∨ Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)

Tactic:
·

Goal state after:
case e_a.e_f.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0 ∨ Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ}, y ∈ Ico 0 1 → HasSum (fun n ↦ 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ k ≠ 1

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case e_a.e_f.inr.inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)

pp: case e_a.e_f.inr.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hΦ' : Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)

Tactic:
·

Goal state after:
case e_a.e_f.inr.inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs : s ≠ 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case e_a.e_f.inr.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hΦ' : Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)

Tactic:
·

Goal state after:
case e_a.e_f.inr.inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hΦ' : Φ 0 = 0
⊢ Φ 0 * hurwitzZetaEven (toAddCircle 0) s = Φ 0 * (completedHurwitzZetaEven (toAddCircle 0) s / s.Gammaℝ)
Module: Mathlib.NumberTheory.LSeries.Linearity

Goal state before:
pp: f : ℕ → ℂ
c s : ℂ
n : ℕ
⊢ term (c • f) s n = c * term f s n

Tactic:
simp [term_smul]

Goal state after:
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
⊢ k ≠ 1

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = 0
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

pp: case inr.inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = 1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

pp: case inr.inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = -1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

Tactic:
·

Goal state after:
case inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = 0
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inr.inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = 1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

pp: case inr.inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = -1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

Tactic:
·

Goal state after:
case inr.inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = 1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inr.inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = -1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x

Tactic:
·

Goal state after:
case inr.inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
p : ℕ
hp : Nat.Prime p
k : ℕ
h : χ ↑p = -1
⊢ 0 ≤ ∑ x ∈ Finset.range (k + 1), χ ↑p ^ x
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ k ≠ 0

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Injectivity

Goal state before:
pp: case inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : m < n + 1
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

pp: case inr.inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
hm : n + 1 ≠ n + 1
⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)

pp: case inr.inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : n + 1 < m
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

Tactic:
·

Goal state after:
case inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : m < n + 1
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
⊢ 0 ≤ χ.zetaMul 0

pp: case inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ χ.zetaMul n

Tactic:
·

Goal state after:
case inl
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
⊢ 0 ≤ χ.zetaMul 0
Module: Mathlib.NumberTheory.LSeries.Injectivity

Goal state before:
pp: case inr.inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
hm : n + 1 ≠ n + 1
⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)

pp: case inr.inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : n + 1 < m
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

Tactic:
·

Goal state after:
case inr.inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
hm : n + 1 ≠ n + 1
⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ χ.zetaMul n

Tactic:
·

Goal state after:
case inr
N : ℕ
χ : DirichletCharacter ℂ N
hχ : χ ^ 2 = 1
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ χ.zetaMul n
Module: Mathlib.NumberTheory.LSeries.Injectivity

Goal state before:
pp: case inr.inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : n + 1 < m
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

Tactic:
·

Goal state after:
case inr.inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
x : ℝ
m : ℕ
hm : m ≠ n + 1
H : n + 1 < m
⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ 1 / ↑n ^ k * (fourier n) ↑y = -(2 * ↑π * I) ^ k / ↑k ! * (-↑k ! / (2 * ↑π * I * ↑n) ^ k) • (fourier n) ↑y

pp: case e'_6
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ ↑(bernoulliFun k y) = B ↑y

Tactic:
·

Goal state after:
case e'_5
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ 1 / ↑n ^ k * (fourier n) ↑y = -(2 * ↑π * I) ^ k / ↑k ! * (-↑k ! / (2 * ↑π * I * ↑n) ^ k) • (fourier n) ↑y
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case h
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ DifferentiableAt ℂ ?f s

pp: case hL
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ B.F =ᶠ[𝓝 s] ?f

pp: case f
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ ℂ → ℂ

Tactic:
·

Goal state after:
case h
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ DifferentiableAt ℂ ?f s
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case hL
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ B.F =ᶠ[𝓝 s] riemannZeta * LFunction B.χ

Tactic:
·

Goal state after:
case hL
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : s ≠ 1
⊢ B.F =ᶠ[𝓝 s] riemannZeta * LFunction B.χ
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
⊢ Function.update (fun s ↦ riemannZeta s * LFunction B.χ s) 1 (deriv (LFunction B.χ) 1) s =
    LSeries (⇑↑zeta) s * LSeries (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s

pp: case hf
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
⊢ LSeriesSummable (⇑↑zeta) s

pp: case hg
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
⊢ LSeriesSummable (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s

Tactic:
·

Goal state after:
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
⊢ Function.update (fun s ↦ riemannZeta s * LFunction B.χ s) 1 (deriv (LFunction B.χ) 1) s =
    LSeries (⇑↑zeta) s * LSeries (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case e'_5
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ (2 * ↑π * I) ^ k ≠ 0

pp: case e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ ↑k ! ≠ 0

Tactic:
·

Goal state after:
case e'_5
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ (2 * ↑π * I) ^ k ≠ 0
Module: Mathlib.NumberTheory.LSeries.Nonvanishing

Goal state before:
pp: case e_a
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ riemannZeta s = LSeries (⇑↑zeta) s

pp: case e_a
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ LSeries (fun x ↦ B.χ ↑x) s = LSeries (⇑(toArithmeticFunction fun x ↦ B.χ ↑x)) s

Tactic:
·

Goal state after:
case e_a
N : ℕ
inst✝ : NeZero N
B : BadChar N
s : ℂ
hs : 1 < s.re
hs' : s ≠ 1
⊢ riemannZeta s = LSeries (⇑↑zeta) s