theorem mellin_zetaKernel₁_eq_tsum {s : ℂ} (hs : 1 / 2 < s.re) :
    mellin zetaKernel₁ s = (π : ℂ) ^ (-s) * Gamma s * ∑' n : ℕ, 1 / ((n : ℂ) + 1) ^ (2 * s) := by
  let bd : ℕ → ℝ → ℝ := fun n t => t ^ (s.re - 1) * exp (-π * t * ((n : ℝ) + 1) ^ 2)
  let f : ℕ → ℝ → ℂ := fun n t => (t : ℂ) ^ (s - 1) * exp (-π * t * ((n : ℝ) + 1) ^ 2)
  have hm : MeasurableSet (Ioi (0 : ℝ)) := measurableSet_Ioi
  have h_norm : ∀ (n : ℕ) {t : ℝ} (_ : 0 < t), ‖f n t‖ = bd n t := by
    intro n t ht
    rw [norm_mul, Complex.norm_eq_abs, Complex.norm_eq_abs, Complex.abs_of_nonneg (exp_pos _).le,
      abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re]
  have hf_meas : ∀ n : ℕ, AEStronglyMeasurable (f n) (volume.restrict <| Ioi 0) := by
    intro n
    refine (ContinuousOn.mul ?_ ?_).aestronglyMeasurable hm
    · exact ContinuousAt.continuousOn fun x hx =>
          continuousAt_ofReal_cpow_const _ _ <| Or.inr <| ne_of_gt hx
    · apply Continuous.continuousOn
      exact continuous_ofReal.comp
          (continuous_exp.comp ((continuous_const.mul continuous_id').mul continuous_const))
  have h_le : ∀ n : ℕ, ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0), ‖f n t‖ ≤ bd n t := fun n =>
    (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => le_of_eq (h_norm n ht))
  have h_sum0 : ∀ {t : ℝ} (_ : 0 < t), HasSum (fun n => f n t)
      ((t : ℂ) ^ (s - 1) * zetaKernel₁ t) := by
    intro t ht
    rw [zetaKernel₁]
    convert
      (hasSum_ofReal.mpr (summable_exp_neg_pi_mul_nat_sq ht).hasSum).mul_left ((t : ℂ) ^ (s - 1))
    simp only [neg_mul, ofReal_exp, ofReal_neg, ofReal_mul, ofReal_pow, ofReal_add,
      ofReal_nat_cast, ofReal_one, ofReal_tsum]
  have h_sum' : ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0), HasSum (fun n : ℕ => f n t)
      ((t : ℂ) ^ (s - 1) * zetaKernel₁ t) :=
    (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => h_sum0 ht)
  have h_sum : ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0), Summable fun n : ℕ => bd n t := by
    refine (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => ?_)
    simpa only [fun n => h_norm n ht] using (h_sum0 ht).summable.norm
  have h_int : Integrable (fun t : ℝ => ∑' n : ℕ, bd n t) (volume.restrict (Ioi 0)) := by
    refine IntegrableOn.congr_fun
        (mellinConvergent_of_isBigO_rpow_exp pi_pos locally_integrable_zetaKernel₁
            isBigO_atTop_zetaKernel₁ isBigO_zero_zetaKernel₁ hs).norm (fun t ht => ?_) hm
    rw [tsum_mul_left, norm_smul, Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos ht, sub_re,
      one_re, zetaKernel₁, ← ofReal_tsum, Complex.norm_eq_abs, Complex.abs_of_nonneg]
    exact tsum_nonneg fun n => (exp_pos _).le
  rw [← tsum_mul_left]
  simp_rw [← integral_cpow_mul_exp_neg_pi_mul_sq (one_half_pos.trans hs)]
  

Proof state:
s : ℂ
hs : 1 / 2 < s.re
bd : ℕ → ℝ → ℝ := fun n t => t ^ (s.re - 1) * rexp (-π * t * (↑n + 1) ^ 2)
f : ℕ → ℝ → ℂ := fun n t => ↑t ^ (s - 1) * ↑(rexp (-π * t * (↑n + 1) ^ 2))
hm : MeasurableSet (Ioi 0)
h_norm : ∀ (n : ℕ) {t : ℝ}, 0 < t → ‖f n t‖ = bd n t
hf_meas : ∀ (n : ℕ), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))
h_le : ∀ (n : ℕ), ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), ‖f n t‖ ≤ bd n t
h_sum0 : ∀ {t : ℝ}, 0 < t → HasSum (fun n => f n t) (↑t ^ (s - 1) * zetaKernel₁ t)
h_sum' : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (↑t ^ (s - 1) * zetaKernel₁ t)
h_sum : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), Summable fun n => bd n t
h_int : Integrable fun t => ∑' (n : ℕ), bd n t
⊢ mellin zetaKernel₁ s = ∑' (x : ℕ), ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) * ↑(rexp (-π * t * (↑x + 1) ^ 2))

Next tactic:
rw [← (hasSum_integral_of_dominated_convergence bd hf_meas h_le h_sum h_int h_sum').tsum_eq.symm]