What question did this study set out to answer?

The aim is to establish an equivalence between the Riemann Hypothesis and a topological statement about zeta zeros and persistence lifetimes.

April 7, 2026Open Access

A Persistence Equivalence for the Riemann Hypothesis (Working Draft)

Key Points

The aim is to establish an equivalence between the Riemann Hypothesis and a topological statement about zeta zeros and persistence lifetimes.
Proves a forward direction using the von Koch estimate and Bottleneck Stability Theorem.
Establishes a reverse direction based on Ingham omega theorems and Fourier attenuation analysis.
Utilizes concepts from polynomial orthogonality and 1D persistent homology.
Demonstrates that nontrivial zeta zeros lie on the critical line if and only if certain conditions about persistence lifetimes are met.
Establishes the relationship L_max(N) = O(√N polylog(N)).

Abstract

We aim to prove that the Riemann Hypothesis is equivalent to a topological statement: all nontrivial zeta zeros lie on the critical line if and only if the maximum persistence lifetime of the prime gravitational manifold satisfies Lₘax (N) = O (√N polylog (N) ). The forward direction chains the von Koch estimate, a Riemann-Stieltjes convolution bound, and the Bottleneck Stability Theorem. The reverse direction uses the Ingham omega theorems, Fourier attenuation analysis, polynomial orthogonality, and the Elder Rule of 1D persistent homology. Working draft v1. 0; formally typeset manuscript in preparation. Fifth paper in the Prime Gravity series.

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