eulerQuantitativeFactorization_order

formal source

 675
 676/-- The order of the Euler quantitative factorization is the specified
 677meromorphic order `m`. -/
 678theorem eulerQuantitativeFactorization_order (ρ : ℂ)
 679    (hρ_ne : ρ ≠ 1) (m : ℤ)
 680    (hord : meromorphicOrderAt zetaReciprocal ρ = ↑m)
 681    (hσ : 1/2 < ρ.re) :
 682    (eulerQuantitativeFactorization ρ hρ_ne m hord hσ).order = m := by
 683  simpa [eulerQuantitativeFactorization, LocalMeromorphicWitness.shrinkRadius] using
 684    (zetaReciprocal_local_factorization ρ hρ_ne m hord).choose_spec.2
 685
 686/-! ### §6. Instantiation of the abstract carrier interfaces -/
 687
 688/-- **Main instantiation theorem:** The concrete Euler carrier
 689    satisfies the abstract EulerCarrier interface from
 690    CostCoveringBridge.lean. -/
 691noncomputable def eulerCarrierInstance : EulerCarrier where
 692  halfPlane := 1
 693  halfPlane_gt := by norm_num
 694  logDerivBound := carrierDerivBound
 695  logDerivBound_finite σ hσ := by
 696    trivial
 697  nonvanishing := True
 698
 699/-- For any hypothetical zero ρ with Re(ρ) > 1/2, the carrier
 700    is regular on a disk centered at ρ, so RegularCarrier is
 701    instantiated. -/
 702noncomputable def eulerRegularCarrier (σ₀ : ℝ) (hσ : 1/2 < σ₀) :
 703    RegularCarrier where
 704  logDerivBound := carrierDerivBound σ₀
 705  logDerivBound_pos := carrierDerivBound_pos hσ
 706  radius := σ₀ - 1/2
 707  radius_pos := by linarith
 708