Cheeger Inequalities for Covering Radii and the Critical Line: A Geometric Analogy for the Riemann Hypothesis

Abstract

This essay speculates on a structural analogy between the covering radius bounds for lattice holes in the Brill-Noether theory of graphs and the distribution of zeros of the Riemann zeta function.

Overview

This article mines the geometry-of-numbers techniques introduced by Manjunath in the study of asymptotic Brill-Noether existence on graphs for potential structural parallels to the Riemann Hypothesis. The source paper establishes a Cheeger-style inequality that bounds the covering radius of a periodic set of "holes" (Crit△(L_G)) with respect to an energy quadratic form, using the spectral gap of the graph. This is applied at the "half-canonical degree" (genus minus one) to prove the existence of divisor classes with rank roughly the square root of the genus.

The Proposed Analogy

We propose a speculative dictionary mapping the discrete geometric structures in the source paper to the analytic theory of the Riemann zeta function ζ(s). The "half-canonical degree" suggests a superficial parallel to the critical line Re(s) = 1/2. The covering radius of the hole set is tentatively mapped to the maximal gap between consecutive zeta zeros, while the spectral gap of the graph is mapped to the mean zero spacing. The energy quadratic form on the Laplacian lattice is analogized to a hypothetical metric governing the "energy" of zero configurations.

Strength of the Correspondence

The correspondence is rated as a suggestive metaphor only. While both contexts involve a "half" parameter (half-canonical vs. critical line) and spectral gap constraints, the source structure relies on finite-dimensional lattices and periodic sets, whereas the zeta zeros form an infinite, non-periodic point process. The analogy is presented as a pattern to be tested rather than a proved equivalence.

Proposed Tests and Failure Modes

We propose experiments comparing the covering radius statistics of random graphs (expander families) to the gap statistics of zeta zeros. If the maximum gap in a graph model grows proportionally to the square root of the number of vertices (mirroring the √g bound in the source), while zeta zero gaps grow only logarithmically, the analogy fails quantitatively. The primary obstruction is the lack of a lattice structure or functional equation analogue on the zeta side that corresponds to the Laplacian lattice energy form.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.