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

Goal state before:
pp: case refine_3.refine_1
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : s ≠ 1
hpc : IsPreconnected {1}ᶜ
hne : 2 ∈ {1}ᶜ
⊢ (fun t ↦ Real.lt✝ 1 t.re) ∈ 𝓝 2

pp: case refine_3.refine_2
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : s ≠ 1
hpc : IsPreconnected {1}ᶜ
hne : 2 ∈ {1}ᶜ
t : ℂ
ht : 1 < t.re
⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))) t

Tactic:
·

Goal state after:
case refine_3.refine_1
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : s ≠ 1
hpc : IsPreconnected {1}ᶜ
hne : 2 ∈ {1}ᶜ
⊢ (fun t ↦ Real.lt✝ 1 t.re) ∈ 𝓝 2
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case refine_3.refine_2
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : s ≠ 1
hpc : IsPreconnected {1}ᶜ
hne : 2 ∈ {1}ᶜ
t : ℂ
ht : 1 < t.re
⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))) t

Tactic:
·

Goal state after:
case refine_3.refine_2
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
hs : s ≠ 1
hpc : IsPreconnected {1}ᶜ
hne : 2 ∈ {1}ᶜ
t : ℂ
ht : 1 < t.re
⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))) t
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : s ≠ 1 ∨ ∑ j, Φ j = 0
hs' : s ≠ 1
⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) s

pp: case inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hs : 1 ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) 1

Tactic:
·

Goal state after:
case inl
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : s ≠ 1 ∨ ∑ j, Φ j = 0
hs' : s ≠ 1
⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) s
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hs : 1 ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) 1

Tactic:
·

Goal state after:
case inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hs : 1 ≠ 1 ∨ ∑ j, Φ j = 0
⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) 1
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inl
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
h : χ ≠ 1
⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))

pp: case inr
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
h : s ≠ 1
⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))

Tactic:
·

Goal state after:
case inl
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
h : χ ≠ 1
⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inr
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
h : s ≠ 1
⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))

Tactic:
·

Goal state after:
case inr
M N : ℕ
inst✝¹ : NeZero M
inst✝ : NeZero N
hMN : M ∣ N
χ : DirichletCharacter ℂ M
s : ℂ
h : s ≠ 1
⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case pos
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
h : j = 0
⊢ IsOpen U

pp: case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
h : ¬j = 0
⊢ IsOpen U

Tactic:
·

Goal state after:
case pos
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
h : j = 0
⊢ IsOpen U
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
h : ¬j = 0
⊢ IsOpen U

Tactic:
·

Goal state after:
case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
h : ¬j = 0
⊢ IsOpen U
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inl
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
s : ℂ
h : s ≠ 0 ∨ N ≠ 1
hχ : χ.Even
⊢ LFunction χ s = completedLFunction χ s / s.Gammaℝ

pp: case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
s : ℂ
h : s ≠ 0 ∨ N ≠ 1
hχ : χ.Odd
⊢ LFunction χ s = completedLFunction χ s / (s + 1).Gammaℝ

Tactic:
·

Goal state after:
case inl
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
s : ℂ
h : s ≠ 0 ∨ N ≠ 1
hχ : χ.Even
⊢ LFunction χ s = completedLFunction χ s / s.Gammaℝ
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
s : ℂ
h : s ≠ 0 ∨ N ≠ 1
hχ : χ.Odd
⊢ LFunction χ s = completedLFunction χ s / (s + 1).Gammaℝ

Tactic:
·

Goal state after:
case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
s : ℂ
h : s ≠ 0 ∨ N ≠ 1
hχ : χ.Odd
⊢ LFunction χ s = completedLFunction χ s / (s + 1).Gammaℝ
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inl
s : ℂ
inst✝ : NeZero 1
χ : DirichletCharacter ℂ 1
hχ : χ.IsPrimitive
⊢ completedLFunction χ (1 - s) = ↑1 ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s

pp: case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s

Tactic:
·

Goal state after:
case inl
s : ℂ
inst✝ : NeZero 1
χ : DirichletCharacter ℂ 1
hχ : χ.IsPrimitive
⊢ completedLFunction χ (1 - s) = ↑1 ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h : χ = 1
⊢ 1 = N

Tactic:
rwa [IsPrimitive, h, conductor_one] at hχ

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

Goal state before:
pp: case pos
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
h : χ.Even
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1) * χ (-1) / I ^ 0 * ZMod.completedLFunction (𝓕 ⇑χ) s

pp: case neg
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
h : ¬χ.Even
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1) * χ (-1) / I ^ 1 * ZMod.completedLFunction (𝓕 ⇑χ) s

Tactic:
·

Goal state after:
case pos
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
h : χ.Even
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1) * χ (-1) / I ^ 0 * ZMod.completedLFunction (𝓕 ⇑χ) s
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case neg
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
h : ¬χ.Even
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1) * χ (-1) / I ^ 1 * ZMod.completedLFunction (𝓕 ⇑χ) s

Tactic:
·

Goal state after:
case neg
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
h : ¬χ.Even
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1) * χ (-1) / I ^ 1 * ZMod.completedLFunction (𝓕 ⇑χ) s
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case pos
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ g U
h : j = 0
⊢ IsPreconnected U

pp: case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ g U
h : ¬j = 0
⊢ IsPreconnected U

Tactic:
·

Goal state after:
case pos
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ g U
h : j = 0
⊢ IsPreconnected U
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ g U
h : ¬j = 0
⊢ IsPreconnected U

Tactic:
·

Goal state after:
case neg
N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ g U
h : ¬j = 0
⊢ IsPreconnected U
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inl
n : ℕ
inst✝ : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
hw : 1 ∈ {s | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
⊢ LFunctionTrivChar₁ n 1 ≠ 0

pp: case inr
n : ℕ
inst✝ : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : w ∈ {s | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
hw' : w ≠ 1
⊢ LFunctionTrivChar₁ n w ≠ 0

Tactic:
·

Goal state after:
case inl
n : ℕ
inst✝ : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
hw : 1 ∈ {s | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
⊢ LFunctionTrivChar₁ n 1 ≠ 0
Module: Mathlib.NumberTheory.LSeries.DirichletContinuation

Goal state before:
pp: case inr
n : ℕ
inst✝ : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : w ∈ {s | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
hw' : w ≠ 1
⊢ LFunctionTrivChar₁ n w ≠ 0

Tactic:
·

Goal state after:
case inr
n : ℕ
inst✝ : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : w ∈ {s | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
hw' : w ≠ 1
⊢ LFunctionTrivChar₁ n w ≠ 0
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))

pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ 1 < (2 * ↑k).re

Tactic:
·

Goal state after:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case refine_1
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k) * Real.cos (2 * π * ↑b * x))

pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))

Tactic:
·

Goal state after:
case refine_1
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k) * Real.cos (2 * π * ↑b * x))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))

Tactic:
·

Goal state after:
case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case pos
k : ℕ
h : k = 1
⊢ (algebraMap ℚ ℝ) (bernoulli' k) = ↑(bernoulli k) + 1

pp: case neg
k : ℕ
h : ¬k = 1
⊢ (algebraMap ℚ ℝ) (bernoulli' k) = ↑(bernoulli k) + 0

Tactic:
·

Goal state after:
case pos
k : ℕ
h : k = 1
⊢ (algebraMap ℚ ℝ) (bernoulli' k) = ↑(bernoulli k) + 1
Module: Mathlib.NumberTheory.KummerDedekind

Goal state before:
pp: case pos.hf
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : Module.IsTorsionFree R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
  ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
    normalizedFactors (Ideal.map (algebraMap R S) I)
this :
  Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
      (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count
      (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
          ⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
      (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
  (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
      ⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
    J'
⊢ Function.Injective fun f ↦ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f)

Tactic:
·

Goal state after:
case pos.hf
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : Module.IsTorsionFree R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
  ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
    normalizedFactors (Ideal.map (algebraMap R S) I)
this :
  Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
      (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count
      (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
          ⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
      (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
  (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
      ⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
    J'
⊢ Function.Injective fun f ↦ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f)
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case pos
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : Φ 0 = 0 ∨ s ≠ 1
j : ZMod N
h : -j ≠ 0 ∨ s ≠ 1
⊢ Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s

pp: case neg
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : Φ 0 = 0 ∨ s ≠ 1
j : ZMod N
h : ¬(-j ≠ 0 ∨ s ≠ 1)
⊢ Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s

Tactic:
·

Goal state after:
case pos
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : Φ 0 = 0 ∨ s ≠ 1
j : ZMod N
h : -j ≠ 0 ∨ s ≠ 1
⊢ Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case neg
k : ℕ
h : ¬k = 1
⊢ (algebraMap ℚ ℝ) (bernoulli' k) = ↑(bernoulli k) + 0

Tactic:
·

Goal state after:
case neg
k : ℕ
h : ¬k = 1
⊢ (algebraMap ℚ ℝ) (bernoulli' k) = ↑(bernoulli k) + 0
Module: Mathlib.NumberTheory.LSeries.ZMod

Goal state before:
pp: case neg
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : Φ 0 = 0 ∨ s ≠ 1
j : ZMod N
h : ¬(-j ≠ 0 ∨ s ≠ 1)
⊢ Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s

Tactic:
·

Goal state after:
case neg
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
s : ℂ
hs : Φ 0 = 0 ∨ s ≠ 1
j : ZMod N
h : ¬(-j ≠ 0 ∨ s ≠ 1)
⊢ Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ 1 < (2 * ↑k).re

Tactic:
·

Goal state after:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ 1 < (2 * ↑k).re
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case hderiv
k : ℕ
x y : ℝ
⊢ ∀ x_1 ∈ [[x, y]], HasDerivAt (bernoulliFun (k + 1)) (↑(k + 1) * bernoulliFun k x_1) x_1

pp: case hint
k : ℕ
x y : ℝ
⊢ IntervalIntegrable (fun t ↦ ↑(k + 1) * bernoulliFun k t) volume x y

Tactic:
·

Goal state after:
case hderiv
k : ℕ
x y : ℝ
⊢ ∀ x_1 ∈ [[x, y]], HasDerivAt (bernoulliFun (k + 1)) (↑(k + 1) * bernoulliFun k x_1) x_1
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case hint
k : ℕ
x y : ℝ
⊢ IntervalIntegrable (fun t ↦ ↑(k + 1) * bernoulliFun k t) volume x y

Tactic:
·

Goal state after:
case hint
k : ℕ
x y : ℝ
⊢ IntervalIntegrable (fun t ↦ ↑(k + 1) * bernoulliFun k t) volume x y
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.sin (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k + 1) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))

pp: k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ 1 < (2 * ↑k + 1).re

Tactic:
·

Goal state after:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.sin (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k + 1) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case refine_1
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.sin (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k + 1) =
    ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k + 1) * Real.sin (2 * π * ↑b * x))

pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))

Tactic:
·

Goal state after:
case refine_1
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ∑' (b : ℕ), ↑(Real.sin (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k + 1) =
    ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k + 1) * Real.sin (2 * π * ↑b * x))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaValues

Goal state before:
pp: case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))

Tactic:
·

Goal state after:
case refine_2
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
        Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) =
    (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
      Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k + 1)))
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case hf
m : ℕ
m0 : m ≠ 0
x✝ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := ⋯
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
⊢ HasDerivAt (fun x ↦ bernoulliFun (k + 1) (↑m * x)) (↑m * (↑k + 1) * bernoulliFun k (↑m * x)) x

pp: case hg
m : ℕ
m0 : m ≠ 0
x✝ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := ⋯
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
⊢ HasDerivAt (fun x ↦ ∑ x_1 ∈ Finset.range m, ↑m ^ k * bernoulliFun (k + 1) (x + ↑x_1 / ↑m))
    (∑ x_1 ∈ Finset.range m, ↑m ^ k * (↑k + 1) * bernoulliFun k (x + ↑x_1 / ↑m)) x

Tactic:
·

Goal state after:
case hf
m : ℕ
m0 : m ≠ 0
x✝ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := ⋯
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
⊢ HasDerivAt (fun x ↦ bernoulliFun (k + 1) (↑m * x)) (↑m * (↑k + 1) * bernoulliFun k (↑m * x)) x
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case hg
m : ℕ
m0 : m ≠ 0
x✝ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := ⋯
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
⊢ HasDerivAt (fun x ↦ ∑ x_1 ∈ Finset.range m, ↑m ^ k * bernoulliFun (k + 1) (x + ↑x_1 / ↑m))
    (∑ x_1 ∈ Finset.range m, ↑m ^ k * (↑k + 1) * bernoulliFun k (x + ↑x_1 / ↑m)) x

Tactic:
·

Goal state after:
case hg
m : ℕ
m0 : m ≠ 0
x✝ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := ⋯
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
⊢ HasDerivAt (fun x ↦ ∑ x_1 ∈ Finset.range m, ↑m ^ k * bernoulliFun (k + 1) (x + ↑x_1 / ↑m))
    (∑ x_1 ∈ Finset.range m, ↑m ^ k * (↑k + 1) * bernoulliFun k (x + ↑x_1 / ↑m)) x
Module: Mathlib.NumberTheory.ZetaValues

Goal state before:
pp: case hg
m : ℕ
m0 : m ≠ 0
x✝¹ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
i : ℕ
x✝ : i ∈ Finset.range m
⊢ HasDerivAt (fun x ↦ ↑m ^ k * bernoulliFun (k + 1) (x + ↑i / ↑m)) (↑m ^ k * (↑k + 1) * bernoulliFun k (x + ↑i / ↑m)) x

Tactic:
simp only [mul_assoc, ← Nat.cast_add_one]

Goal state after:
case hg
m : ℕ
m0 : m ≠ 0
x✝¹ : ℝ
m0' : ↑m ≠ 0
f : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)
k : ℕ
h : ∀ (x : ℝ), f k x = 0
x : ℝ
i : ℕ
x✝ : i ∈ Finset.range m
⊢ HasDerivAt (fun x ↦ ↑m ^ k * bernoulliFun (k + 1) (x + ↑i / ↑m)) (↑m ^ k * (↑(k + 1) * bernoulliFun k (x + ↑i / ↑m)))
    x