have := (residueAtNegOne xpos).and <| verticalIntegral_sub_verticalIntegral_eq_squareIntegral
  (by simpa using ⟨by linarith, by linarith⟩) hf
  (tendsto_zero_Lower xpos _ _) (tendsto_zero_Upper xpos _ _)
  (isIntegrable xpos (by norm_num) (by norm_num)) (isIntegrable xpos (by norm_num) (by norm_num))
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
x_gt_one : 1 < x
xpos : 0 < x
hf : HolomorphicOn (f x) (Icc (-3 / 2) (-1 / 2) ×ℂ univ \ {-1})
⊢ VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-1 / 2) -
      VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-3 / 2) =
    -1 / ↑x
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
x_gt_one : 1 < x
xpos : 0 < x
hf : HolomorphicOn (f x) (Icc (-3 / 2) (-1 / 2) ×ℂ univ \ {-1})
this :
  ∀ᶠ (x_1 : ℝ) in 𝓝[>] 0,
    RectangleIntegral' (f x) (-↑x_1 - ↑x_1 * I - 1) (↑x_1 + ↑x_1 * I - 1) = -↑x⁻¹ ∧
      VerticalIntegral (f x) (-1 / 2) - VerticalIntegral (f x) (-3 / 2) =
        RectangleIntegral (f x) (-↑x_1 - ↑x_1 * I + -1) (↑x_1 + ↑x_1 * I + -1)
⊢ VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-1 / 2) -
      VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-3 / 2) =
    -1 / ↑x