chebyshev_asymptotic'

theorem chebyshev_asymptotic' :    ∃ (f : ℝ → ℝ),      (∀ ε > (0 : ℝ), (f =o[atTop] fun t ↦ ε * t)) ∧      (∀ (x : ℝ), 2 ≤ x → IntegrableOn f (Set.Icc 2 x)) ∧      ∀ (x : ℝ), θ x = x + f x := by  have H := chebyshev_asymptotic  rw [IsEquivalent, isLittleO_iff] at H  let f := (fun x ↦ θ x - x)  have integrable (x : ℝ) (hx : 2 ≤ x) : IntegrableOn f (Set.Icc 2 x) := by    rw [IntegrableOn]    refine Integrable.sub ?_ (ContinuousOn.integrableOn_Icc (continuousOn_id' _))    refine Chebyshev.integrableOn_theta_div_id_mul_log_sq x |>.mul_continuousOn (g' := fun t => t * log t ^ 2)      (ContinuousOn.mul (continuousOn_id' _) (ContinuousOn.pow (continuousOn_log |>.mono <| by        rintro t ⟨ht1, _⟩        simp only [Set.mem_compl_iff, Set.mem_singleton_iff]        linarith) 2)) isCompact_Icc |>.congr_fun_ae ?_    simp only [measurableSet_Icc, ae_restrict_eq, EventuallyEq, eventually_inf_principal]    refine .of_forall fun t ⟨ht1, _⟩ => ?_    rw [div_mul_cancel₀]    simpa only [ne_eq, _root_.mul_eq_zero, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff,      log_eq_zero, or_self_left, not_or] using ⟨by linarith, by linarith, by linarith⟩  refine ⟨f, fun ε hε ↦ ?_, integrable, ?_⟩  · rw [isLittleO_iff]    intro c hc    specialize @H (c * ε) (mul_pos hc hε)    simp only [Pi.sub_apply, norm_eq_abs, mul_assoc, eventually_atTop, norm_mul,      abs_of_pos hε, f] at H ⊢    exact H  refine fun r => by simp [f]