eulerQuantitativeFactorization_logDerivBound

formal source

 655
 656/-- The log-derivative bound of the Euler quantitative factorization
 657equals the carrier derivative bound at `σ₀ = Re(ρ)`. -/
 658theorem eulerQuantitativeFactorization_logDerivBound (ρ : ℂ)
 659    (hρ_ne : ρ ≠ 1) (m : ℤ)
 660    (hord : meromorphicOrderAt zetaReciprocal ρ = ↑m)
 661    (hσ : 1/2 < ρ.re) :
 662    (eulerQuantitativeFactorization ρ hρ_ne m hord hσ).logDerivBound =
 663      carrierDerivBound ρ.re := by
 664  simp [eulerQuantitativeFactorization]
 665
 666/-- The center of the Euler quantitative factorization is the actual complex
 667point `ρ`. -/
 668theorem eulerQuantitativeFactorization_center (ρ : ℂ)
 669    (hρ_ne : ρ ≠ 1) (m : ℤ)
 670    (hord : meromorphicOrderAt zetaReciprocal ρ = ↑m)
 671    (hσ : 1/2 < ρ.re) :
 672    (eulerQuantitativeFactorization ρ hρ_ne m hord hσ).center = ρ := by
 673  simpa [eulerQuantitativeFactorization, LocalMeromorphicWitness.shrinkRadius] using
 674    (zetaReciprocal_local_factorization ρ hρ_ne m hord).choose_spec.1
 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