Liouville Parity Holomorphy Criterion

Abstract

Holomorphy of the Liouville parity quotient characterizes the zeta zero line.

Theorem 1.1 (The Liouville parity quotient is holomorphic exactly on the zero-line criterion).

$$\begin{array}{r}\mathrm{let}observationHalfPlane:\mathrm{Set}\left(\mathbb{C}\right):=\{s\in \mathbb{C}\mid \frac{1}{2}<\Re (s)\},\\ \mathrm{let}liouvilleParity:\mathbb{C}\to \mathbb{C}:=(s:\mathbb{C}\mapsto \frac{\mathrm{riemannZeta}\left(2\times s\right)}{\mathrm{riemannZeta}\left(s\right)}),\\ \mathrm{let}hasHolomorphicParity:\mathrm{Prop}:=\forall s\in \mathbb{C},\mathrm{Mem}\left(s,observationHalfPlane\right)\Rightarrow \left(\exists germ\in \mathbb{C}\to \mathbb{C},\mathrm{AnalyticAt}\left(\mathbb{C},germ,s\right)\wedge \mathrm{EventuallyEq}\left(\mathrm{nhdsWithin}\left(s,\mathbb{C}\setminus \{s\}\right),liouvilleParity,germ\right)\right),\\ \mathrm{RiemannHypothesis}\left(\right)\iff hasHolomorphicParity.\end{array}$$

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

Source. Repository-derived.

Commentary.

The observation region is the open half-plane to the right of one half. Holomorphy means that the literal quotient agrees on each punctured neighborhood with a local analytic germ, so the value assigned at an apparent singularity cannot hide a pole.

The Riemann hypothesis removes denominator zeros from the open half-plane, while the zeta residue factorization supplies an analytic germ at one. Conversely, an off-line zero contributes positive denominator multiplicity while the doubled numerator is nonzero, contradicting analyticity of the local germ.

References

• Truth anchor: D5/S3/Analytic/ZetaObservation/LiouvilleParityHolomorphyCriterion.liouville_parity_holomorphy_criterion
• Dependency: D5/S3/Weil/ZetaBridge/RightHalfStripRiemannReduction