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2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.20025132 |
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| _version_ | 1866902119034912768 |
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| author | Louro, Fernando J. O. |
| author_facet | Louro, Fernando J. O. |
| contents | <p>The Riemann zeta function \zeta(s)=\sum_{n=1}^{\infty} n^{-s} (for \Re(s)>1) and its non-trivial zeros \rho=\sigma+i\gamma have been central to analytic number theory since Riemann's 1859 memoir. The Riemann Hypothesis (RH) asserts that all non-trivial zeros satisfy \sigma=1/2. Despite overwhelming numerical evidence [Odlyzko2001] and deep connections to the distribution of prime numbers via the explicit formula [Riemann1859, Mangoldt, Edwards1974], a general proof has remained elusive for over 165 years.</p> <p> </p> <p>In a series of preceding works [FMF, HENCORA, SCDv5], we introduced the Structural Observational Inversion (SOI) framework, a novel approach that treats the RH not as an isolated analytic conjecture but as a consequence of a fundamental symmetry of the multiplicative vacuum. The core object is the inversion operator \mathcal{J} acting on the Hilbert space L^2(\mathbb{R}_+,dx/x), which forces the critical line \sigma=1/2 as the unique locus of exact covariance. The SOI framework rests on four operational layers:</p> <p> </p> <p>· Helix: the truncated Dirichlet series, representing the oscillatory excitation of the vacuum.</p> <p>· Glue: the inversion operator \mathcal{J} and its associated potential C_\varepsilon(\sigma)=\kappa_\varepsilon(\sigma-1/2)^2, encoding structural memory.</p> <p>· SDC: the Spectral Dynamic Cascade metric M(t)=|\Im(S'_N/S_N)|, which detects phase singularities at the zeros.</p> <p>· Prima: the local covariance filter that separates genuine structural points from transient noise.</p> <p> </p> <p>In the present paper, we develop the continuous-field formulation underlying the SOI framework. We show that the truncated Dirichlet sum can be approximated by a logarithmic integral that evolves according to a simple advection equation. This continuous description explains, without adjustable parameters, the key numerical observations reported in our earlier studies:</p> <p> </p> <p>1. The phase-derivative profile near a zeta zero follows a generalized Lorentzian with exponent p>1.</p> <p>2. The width \delta of this profile scales universally as 1/\log N, where N is the truncation order.</p> <p>3. The SDC metric collapses to zero at the zeros of arithmetic L-functions while remaining of order unity elsewhere, providing a separation of three to four orders of magnitude.</p> <p>4. The Davenport–Heilbronn function, which satisfies a functional equation but lacks an Euler product, fails the SDC test entirely.</p> <p> </p> <p>Furthermore, the continuous formalism makes the connection to the inversion operator \mathcal{J} transparent: in the limit of infinite truncation (L\to\infty), the field becomes symmetric under x\mapsto -x in logarithmic space, and the critical line \sigma=1/2 emerges as the unique fixed point of this symmetry.</p> <p> </p> <p>The paper is organized as follows. Section 2 derives the continuous field equation from the truncated Dirichlet sum. Section 3 computes the phase derivative and analyzes the generalized Lorentzian profile. Section 4 proves the 1/\log N scaling law. Section 5 connects the continuous field to the SOI inversion operator. Section 6 summarizes the numerical evidence, including prime enrichment, Davenport–Heilbronn rejection, and GUE statistics of the partition cascade. Section 7 discusses implications and outlines the path toward a rigorous proof of the RH.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20025132 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Continuous Field of Zeros Ground Louro, Fernando J. O. <p>The Riemann zeta function \zeta(s)=\sum_{n=1}^{\infty} n^{-s} (for \Re(s)>1) and its non-trivial zeros \rho=\sigma+i\gamma have been central to analytic number theory since Riemann's 1859 memoir. The Riemann Hypothesis (RH) asserts that all non-trivial zeros satisfy \sigma=1/2. Despite overwhelming numerical evidence [Odlyzko2001] and deep connections to the distribution of prime numbers via the explicit formula [Riemann1859, Mangoldt, Edwards1974], a general proof has remained elusive for over 165 years.</p> <p> </p> <p>In a series of preceding works [FMF, HENCORA, SCDv5], we introduced the Structural Observational Inversion (SOI) framework, a novel approach that treats the RH not as an isolated analytic conjecture but as a consequence of a fundamental symmetry of the multiplicative vacuum. The core object is the inversion operator \mathcal{J} acting on the Hilbert space L^2(\mathbb{R}_+,dx/x), which forces the critical line \sigma=1/2 as the unique locus of exact covariance. The SOI framework rests on four operational layers:</p> <p> </p> <p>· Helix: the truncated Dirichlet series, representing the oscillatory excitation of the vacuum.</p> <p>· Glue: the inversion operator \mathcal{J} and its associated potential C_\varepsilon(\sigma)=\kappa_\varepsilon(\sigma-1/2)^2, encoding structural memory.</p> <p>· SDC: the Spectral Dynamic Cascade metric M(t)=|\Im(S'_N/S_N)|, which detects phase singularities at the zeros.</p> <p>· Prima: the local covariance filter that separates genuine structural points from transient noise.</p> <p> </p> <p>In the present paper, we develop the continuous-field formulation underlying the SOI framework. We show that the truncated Dirichlet sum can be approximated by a logarithmic integral that evolves according to a simple advection equation. This continuous description explains, without adjustable parameters, the key numerical observations reported in our earlier studies:</p> <p> </p> <p>1. The phase-derivative profile near a zeta zero follows a generalized Lorentzian with exponent p>1.</p> <p>2. The width \delta of this profile scales universally as 1/\log N, where N is the truncation order.</p> <p>3. The SDC metric collapses to zero at the zeros of arithmetic L-functions while remaining of order unity elsewhere, providing a separation of three to four orders of magnitude.</p> <p>4. The Davenport–Heilbronn function, which satisfies a functional equation but lacks an Euler product, fails the SDC test entirely.</p> <p> </p> <p>Furthermore, the continuous formalism makes the connection to the inversion operator \mathcal{J} transparent: in the limit of infinite truncation (L\to\infty), the field becomes symmetric under x\mapsto -x in logarithmic space, and the critical line \sigma=1/2 emerges as the unique fixed point of this symmetry.</p> <p> </p> <p>The paper is organized as follows. Section 2 derives the continuous field equation from the truncated Dirichlet sum. Section 3 computes the phase derivative and analyzes the generalized Lorentzian profile. Section 4 proves the 1/\log N scaling law. Section 5 connects the continuous field to the SOI inversion operator. Section 6 summarizes the numerical evidence, including prime enrichment, Davenport–Heilbronn rejection, and GUE statistics of the partition cascade. Section 7 discusses implications and outlines the path toward a rigorous proof of the RH.</p> |
| title | Continuous Field of Zeros Ground |
| url | https://doi.org/10.5281/zenodo.20025132 |