The Critical Circle: a Topological Cartography of the Critical Line via the Canonical Logarithm
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901498436255744 |
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| author | Hernandez Reveles, Ricardo |
| author_facet | Hernandez Reveles, Ricardo |
| contents | <p>Starting from a single operation — the polar decomposition of the canonical logarithm applied to the Euler product — we derive the topological structure of the critical line of the Riemann zeta function in the <em>b</em>-plane of the specialization map Φ: <em>b</em> ↦ <em>p</em><sup>−<em>s</em></sup> introduced in Paper 1. The unconditional content — tori at every depth, phase coordinates, incommensurability, density, and the strict complexity of zeros — follows from one theorem whose only inputs are the canonical logarithm, the Fundamental Theorem of Arithmetic, and the Kronecker–Weyl equidistribution theorem. The conditional content — the critical circles, the Bohr torus at σ = 1/2, the supercritical/subcritical partition at |<em>b</em>| = 1/2, the spectral compression, and the phase-suppression mechanism — requires one additional datum: the involution ρ(<em>s</em>) = 1 − <em>s</em> from Riemann's functional equation, which depends on the archimedean factor Λ<sub>∞</sub>(<em>s</em>) = π<sup>−<em>s</em>/2</sup> Γ(<em>s</em>/2). This factor is identified as the unique obstruction to closing the chain from the elementary traversals of Paper 1 to the functional equation. Five independent appearances of 1/2 are documented and classified by epistemic status.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19965225 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Critical Circle: a Topological Cartography of the Critical Line via the Canonical Logarithm Hernandez Reveles, Ricardo Riemann zeta function critical line Kronecker-Weyl equidistribution Bohr compactification Euler embedding topological number theory 11M06 — ζ(s) and L(s, χ) 11M26 — Nonreal zeros; Riemann hypothesis <p>Starting from a single operation — the polar decomposition of the canonical logarithm applied to the Euler product — we derive the topological structure of the critical line of the Riemann zeta function in the <em>b</em>-plane of the specialization map Φ: <em>b</em> ↦ <em>p</em><sup>−<em>s</em></sup> introduced in Paper 1. The unconditional content — tori at every depth, phase coordinates, incommensurability, density, and the strict complexity of zeros — follows from one theorem whose only inputs are the canonical logarithm, the Fundamental Theorem of Arithmetic, and the Kronecker–Weyl equidistribution theorem. The conditional content — the critical circles, the Bohr torus at σ = 1/2, the supercritical/subcritical partition at |<em>b</em>| = 1/2, the spectral compression, and the phase-suppression mechanism — requires one additional datum: the involution ρ(<em>s</em>) = 1 − <em>s</em> from Riemann's functional equation, which depends on the archimedean factor Λ<sub>∞</sub>(<em>s</em>) = π<sup>−<em>s</em>/2</sup> Γ(<em>s</em>/2). This factor is identified as the unique obstruction to closing the chain from the elementary traversals of Paper 1 to the functional equation. Five independent appearances of 1/2 are documented and classified by epistemic status.</p> |
| title | The Critical Circle: a Topological Cartography of the Critical Line via the Canonical Logarithm |
| topic | Riemann zeta function critical line Kronecker-Weyl equidistribution Bohr compactification Euler embedding topological number theory 11M06 — ζ(s) and L(s, χ) 11M26 — Nonreal zeros; Riemann hypothesis |
| url | https://doi.org/10.5281/zenodo.19965225 |