The Riemann $\Xi$-function from primitive Markovian cycles II: Strip rigidity and divisor identification

TL;DR

This paper establishes a connection between the Riemann $\Xi$-function and a canonical reference family using a rigidity lemma, showing they share the same zero divisor on certain strips under specific conditions.

Contribution

It introduces a rigidity lemma for holomorphic functions on strips and applies it to relate the $\Xi$-function to a reference family, advancing understanding of their zero distributions.

Findings

$\Xi$-function and reference family share zero divisor on overlap strips

Rigidity lemma ensures zero-free holomorphic extension of the seam ratio

Conditions verified in subsequent work confirm the main hypotheses

Abstract

We compare the Riemann $\Xi$--function to a canonical real-entire reference family arising from the cycle Laplacian developed in Paper I. These spectral determinants have only real zeros by self-adjointness. Our main tool is a rigidity lemma for holomorphic functions on horizontal strips. Applied to a normalized seam ratio linking $\Xi(2\cdot)$ to the reference family, this lemma shows that, under explicit holomorphy and boundary nonvanishing hypotheses verified in the forthcoming Paper III, the seam ratio extends to a zero-free holomorphic function of bounded type on each overlap strip. It follows that, on every admissible overlap strip, $\Xi(2\cdot)$ and the reference family have the same zero divisor.