simp_rw [Asymptotics.isBigO_iff] at this
case calc_2.e_a.e_a
ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 ψ
ψpos : ∀ (x : ℝ), 0 ≤ ψ x
suppΨ : support ψ ⊆ Icc (1 / 2) 2
mass_one : ∫ (x : ℝ) in Ioi 0, ψ x / x = 1
X : ℝ
X_pos : 0 < X
ε : ℝ
εpos : 0 < ε
this : (fun s => ‖𝓜 (fun x => ↑(Smooth1 ψ ε x)) s‖) =O[𝓟 {s | 2 ≤ s.re ∧ s.re ≤ 2}] fun s => 1 / (ε * ‖s‖ ^ 2)
⊢ ∫ (t : ℝ), ∑' (n : ℕ), ↑(Λ n) / ↑n ^ (2 + ↑t * I) * 𝓜 (fun x => ↑(Smooth1 ψ ε x)) (2 + ↑t * I) * ↑X ^ (2 + ↑t * I) =
    ∑' (n : ℕ), ∫ (t : ℝ), ↑(Λ n) / ↑n ^ (2 + ↑t * I) * 𝓜 (fun x => ↑(Smooth1 ψ ε x)) (2 + ↑t * I) * ↑X ^ (2 + ↑t * I)
case calc_2.e_a.e_a
ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 ψ
ψpos : ∀ (x : ℝ), 0 ≤ ψ x
suppΨ : support ψ ⊆ Icc (1 / 2) 2
mass_one : ∫ (x : ℝ) in Ioi 0, ψ x / x = 1
X : ℝ
X_pos : 0 < X
ε : ℝ
εpos : 0 < ε
this : ∃ c, ∀ᶠ (x : ℂ) in 𝓟 {s | 2 ≤ s.re ∧ s.re ≤ 2}, ‖‖𝓜 (fun x => ↑(Smooth1 ψ ε x)) x‖‖ ≤ c * ‖1 / (ε * ‖x‖ ^ 2)‖
⊢ ∫ (t : ℝ), ∑' (n : ℕ), ↑(Λ n) / ↑n ^ (2 + ↑t * I) * 𝓜 (fun x => ↑(Smooth1 ψ ε x)) (2 + ↑t * I) * ↑X ^ (2 + ↑t * I) =
    ∑' (n : ℕ), ∫ (t : ℝ), ↑(Λ n) / ↑n ^ (2 + ↑t * I) * 𝓜 (fun x => ↑(Smooth1 ψ ε x)) (2 + ↑t * I) * ↑X ^ (2 + ↑t * I)