Parity Polarization Pole Orders

Abstract

The parity quotient criterion carries the exact orders of all three observers.

Theorem 1.1 (Parity holomorphy and the three observer pole orders).

$$\begin{array}{r}\mathrm{let}observationHalfPlane:\mathrm{Set}\left(\mathbb{C}\right):=\{s\in \mathbb{C}\mid \frac{1}{2}<\Re (s)\},\\ \mathrm{let}parityPolarization:\mathbb{C}\to \mathbb{C}:=(s:\mathbb{C}\mapsto \frac{\mathrm{riemannZeta}\left(2\times s\right)}{\mathrm{riemannZeta}{\left(s\right)}^2}),\\ \mathrm{let}mobiusObserver:\mathbb{C}\to \mathbb{C}:=(s:\mathbb{C}\mapsto \mathrm{riemannZeta}{\left(s\right)}^{-1}),\\ \mathrm{let}liouvilleObserver:\mathbb{C}\to \mathbb{C}:=(s:\mathbb{C}\mapsto \frac{\mathrm{riemannZeta}\left(2\times s\right)}{\mathrm{riemannZeta}\left(s\right)}),\\ \mathrm{let}hasHolomorphicPolarization:\mathrm{Prop}:=\forall s\in \mathbb{C},\mathrm{Mem}\left(s,observationHalfPlane\right)\Rightarrow \left(\exists germ\in \mathbb{C}\to \mathbb{C},\mathrm{AnalyticAt}\left(\mathbb{C},germ,s\right)\wedge \mathrm{EventuallyEq}\left(\mathrm{nhdsWithin}\left(s,\mathbb{C}\setminus \{s\}\right),parityPolarization,germ\right)\right),\\ \left(\mathrm{RiemannHypothesis}\left(\right)\iff hasHolomorphicPolarization\right)\wedge \left(\forall rho\in \mathbb{C},multiplicity\in \mathbb{N},\left(\mathrm{Mem}\left(rho,observationHalfPlane\right)\wedge \left(\mathrm{riemannZeta}\left(rho\right)=0\wedge \mathrm{zeroMult}\left(rho\right)=multiplicity\right)\right)\Rightarrow \left(\mathrm{meromorphicOrderAt}\left(mobiusObserver,rho\right)=-multiplicity\wedge \left(\mathrm{meromorphicOrderAt}\left(liouvilleObserver,rho\right)=-multiplicity\wedge \mathrm{meromorphicOrderAt}\left(parityPolarization,rho\right)=-2\times multiplicity\right)\right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaObservation/ParityPolarizationPoleOrders.parity_polarization_holomorphy_and_pole_orders (✓ std3). ∎

Source. Repository-derived.

Commentary.

The normalized parity polarization is holomorphic throughout the open observation half-plane exactly when the Riemann hypothesis holds.

At a zeta zero in that half-plane, doubling moves the numerator into the zero-free half-plane. Meromorphic-order subtraction then gives the multiplicity orders for the reciprocal and Liouville observers and twice that order for the normalized polarization.

References

• Truth anchor: D5/S3/Analytic/ZetaObservation/ParityPolarizationPoleOrders.parity_polarization_holomorphy_and_pole_orders
• Dependency: D5/S3/Analytic/ZetaObservation/ParityPolarizationHolomorphyCriterion