Symmetry-Based Boundary Approximation and Zero Localization for the Riemann Zeta Function
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2026
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| _version_ | 1866902085331582976 |
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| author | Fradkin, Yuval |
| author_facet | Fradkin, Yuval |
| contents | <div> <div> <div> <div dir="auto"> <div> <div> </div> </div> </div> </div> <div> </div> <div> <div> <h2>Preface: A Developing Research Program</h2> <p>This article develops a symmetry-based analytic approach to the Riemann Hypothesis that couples dyadic (base-2) structural symmetry with quantitative boundary approximation on narrow rectangles surrounding the critical line. The central analytic mechanism constructs exactly symmetric Dirichlet filters and represents the logarithmic derivative of the Riemann zeta function in the form</p> <span><span><span>F(s)=(−ζ′ζ)(s) G(s),F(s)=\left(-\frac{\zeta'}{\zeta}\right)(s)\,G(s),</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span><span>=</span></span><span><span><span><span>(</span></span><span>−</span><span><span><span><span><span><span>ζ</span><span>ζ</span><span><span><span><span>′</span></span></span></span></span><span></span></span></span></span></span><span><span>)</span></span></span><span>(</span><span>s</span><span>)</span><span>G</span><span>(</span><span>s</span><span>)</span><span>,</span></span></span></span></span> <p>so that zeros of <span><span>ζ(s)\zeta(s)</span><span><span><span>ζ</span><span>(</span><span>s</span><span>)</span></span></span></span> can be detected without introducing artificial cancellation. Within this framework, Abel summation, explicit-formula identities, Littlewood’s rectangle lemma, the Poisson–Jensen formula, and three-lines propagation are combined to convert small symmetric boundary error of <span><span>F(s)F(s)</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span></span></span></span> into interior localization of zeros. This yields global <span><span>O(logT)O(\log T)</span><span><span><span>O</span><span>(</span><span>log</span><span>T</span><span>)</span></span></span></span> control of weighted zero sums and provides a direct mechanism linking boundary approximation to critical-line concentration.</p> <p>The analytic construction employs height-adaptive symmetric Dirichlet filters that approximate <span><span>(−ζ′/ζ)−1(-\zeta'/\zeta)^{-1}</span><span><span><span>(</span><span>−</span><span><span>ζ</span><span><span><span><span><span><span>′</span></span></span></span></span></span></span><span>/</span><span>ζ</span><span>)<span><span><span><span><span><span>−1</span></span></span></span></span></span></span></span></span></span> along the boundary of the rectangle with calibrated accuracy and polynomial complexity. When the boundary approximation is sufficiently precise, the Poisson–Jensen mechanism forces the absence of off-critical zeros in the interior region under mild strip bounds.</p> <p>The present work is written as a <strong>developing research program</strong>. Earlier sections introduce structural definitions and analytic tools that are progressively refined in later sections. As the series advances, technical gaps and heuristic arguments appearing in early parts are revisited and resolved in subsequent articles. Consequently, the paper should be read as a continuous development in which the analytic framework, filter construction, and boundary-to-interior arguments gradually converge toward the final result.</p> <p>The analysis presented here builds upon two foundational works that establish the broader conceptual and analytic framework underlying the present program:</p> <ul> <li> <p><strong>From Energy Alone to All Physics</strong><br><a href="https://zenodo.org/records/18738256" target="_new" rel="noopener">https://zenodo.org/records/18738256</a></p> </li> <li> <p><strong>FEATAM — From Energy Alone to All Mathematics</strong><br><a href="https://zenodo.org/records/18774367" target="_new" rel="noopener">https://zenodo.org/records/18774367</a></p> </li> </ul> <p>These works introduce the structural and energetic perspective that motivates the analytic formulation used in the present study. In particular, the interpretation of analytic approximation as an energy-minimization process within an appropriate Hilbert space framework informs the construction of symmetric Dirichlet filters developed throughout this article.</p> <p>Taken together, the series aims to present a transparent and testable analytic pathway from symmetric boundary approximation to the concentration of zeros on the critical line, with constants and estimates tracked explicitly to allow rigorous scrutiny of each step.</p> </div> </div> </div> </div> <div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19044603 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Symmetry-Based Boundary Approximation and Zero Localization for the Riemann Zeta Function Fradkin, Yuval <div> <div> <div> <div dir="auto"> <div> <div> </div> </div> </div> </div> <div> </div> <div> <div> <h2>Preface: A Developing Research Program</h2> <p>This article develops a symmetry-based analytic approach to the Riemann Hypothesis that couples dyadic (base-2) structural symmetry with quantitative boundary approximation on narrow rectangles surrounding the critical line. The central analytic mechanism constructs exactly symmetric Dirichlet filters and represents the logarithmic derivative of the Riemann zeta function in the form</p> <span><span><span>F(s)=(−ζ′ζ)(s) G(s),F(s)=\left(-\frac{\zeta'}{\zeta}\right)(s)\,G(s),</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span><span>=</span></span><span><span><span><span>(</span></span><span>−</span><span><span><span><span><span><span>ζ</span><span>ζ</span><span><span><span><span>′</span></span></span></span></span><span></span></span></span></span></span><span><span>)</span></span></span><span>(</span><span>s</span><span>)</span><span>G</span><span>(</span><span>s</span><span>)</span><span>,</span></span></span></span></span> <p>so that zeros of <span><span>ζ(s)\zeta(s)</span><span><span><span>ζ</span><span>(</span><span>s</span><span>)</span></span></span></span> can be detected without introducing artificial cancellation. Within this framework, Abel summation, explicit-formula identities, Littlewood’s rectangle lemma, the Poisson–Jensen formula, and three-lines propagation are combined to convert small symmetric boundary error of <span><span>F(s)F(s)</span><span><span><span>F</span><span>(</span><span>s</span><span>)</span></span></span></span> into interior localization of zeros. This yields global <span><span>O(logT)O(\log T)</span><span><span><span>O</span><span>(</span><span>log</span><span>T</span><span>)</span></span></span></span> control of weighted zero sums and provides a direct mechanism linking boundary approximation to critical-line concentration.</p> <p>The analytic construction employs height-adaptive symmetric Dirichlet filters that approximate <span><span>(−ζ′/ζ)−1(-\zeta'/\zeta)^{-1}</span><span><span><span>(</span><span>−</span><span><span>ζ</span><span><span><span><span><span><span>′</span></span></span></span></span></span></span><span>/</span><span>ζ</span><span>)<span><span><span><span><span><span>−1</span></span></span></span></span></span></span></span></span></span> along the boundary of the rectangle with calibrated accuracy and polynomial complexity. When the boundary approximation is sufficiently precise, the Poisson–Jensen mechanism forces the absence of off-critical zeros in the interior region under mild strip bounds.</p> <p>The present work is written as a <strong>developing research program</strong>. Earlier sections introduce structural definitions and analytic tools that are progressively refined in later sections. As the series advances, technical gaps and heuristic arguments appearing in early parts are revisited and resolved in subsequent articles. Consequently, the paper should be read as a continuous development in which the analytic framework, filter construction, and boundary-to-interior arguments gradually converge toward the final result.</p> <p>The analysis presented here builds upon two foundational works that establish the broader conceptual and analytic framework underlying the present program:</p> <ul> <li> <p><strong>From Energy Alone to All Physics</strong><br><a href="https://zenodo.org/records/18738256" target="_new" rel="noopener">https://zenodo.org/records/18738256</a></p> </li> <li> <p><strong>FEATAM — From Energy Alone to All Mathematics</strong><br><a href="https://zenodo.org/records/18774367" target="_new" rel="noopener">https://zenodo.org/records/18774367</a></p> </li> </ul> <p>These works introduce the structural and energetic perspective that motivates the analytic formulation used in the present study. In particular, the interpretation of analytic approximation as an energy-minimization process within an appropriate Hilbert space framework informs the construction of symmetric Dirichlet filters developed throughout this article.</p> <p>Taken together, the series aims to present a transparent and testable analytic pathway from symmetric boundary approximation to the concentration of zeros on the critical line, with constants and estimates tracked explicitly to allow rigorous scrutiny of each step.</p> </div> </div> </div> </div> <div> </div> |
| title | Symmetry-Based Boundary Approximation and Zero Localization for the Riemann Zeta Function |
| url | https://doi.org/10.5281/zenodo.19044603 |