Gorde:
| Egile nagusia: | |
|---|---|
| Formatua: | Recurso digital |
| Hizkuntza: | ingelesa |
| Argitaratua: |
Zenodo
2025
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| Gaiak: | |
| Sarrera elektronikoa: | https://doi.org/10.5281/zenodo.16422652 |
| Etiketak: |
Etiketa erantsi
Etiketarik gabe, Izan zaitez lehena erregistro honi etiketa jartzen!
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Aurkibidea:
- <p>We present a constructive spectral framework for the Riemann zeta function <span><span>ζ(s)\zeta(s)</span><span><span><span>ζ</span><span>(</span><span>s</span><span>)</span></span></span></span>, defining a compact, self-adjoint operator <span><span>T^phys\hat{T}_{\text{phys}}</span><span><span><span><span><span><span><span><span>T</span><span>^</span></span></span></span></span><span><span><span><span><span><span><span>phys</span></span></span></span><span></span></span></span></span></span></span></span></span> on <span><span>ℓ2(N)\ell^2(\mathbb{N})</span><span><span><span>ℓ<span><span><span><span><span><span>2</span></span></span></span></span></span></span><span>(</span><span>N</span><span>)</span></span></span></span>, derived from a physically motivated quantum graph with logarithmic potential. The spectrum of <span><span>T^phys\hat{T}_{\text{phys}}</span><span><span><span><span><span><span><span><span>T</span><span>^</span></span></span></span></span><span><span><span><span><span><span><span>phys</span></span></span></span><span></span></span></span></span></span></span></span></span> is shown to asymptotically match the imaginary parts <span><span>γn\gamma_n</span><span><span><span><span>γ</span><span><span><span><span><span><span>n</span></span></span><span></span></span></span></span></span></span></span></span> of the non-trivial zeros of <span><span>ζ(s)\zeta(s)</span><span><span><span>ζ</span><span>(</span><span>s</span><span>)</span></span></span></span>.</p> <p>We formulate and refine the identity:</p> <p><span><span><span>ζ(s)=detζ(s−T^)⋅eP(s),\zeta(s) = \det\nolimits_\zeta(s - \hat{T}) \cdot e^{P(s)},</span><span><span><span>ζ</span><span>(</span><span>s</span><span>)</span><span>=</span></span><span><span>det<span><span><span><span><span><span>ζ</span></span></span><span></span></span></span></span></span><span>(</span><span>s</span><span>−</span></span><span><span><span><span><span><span>T</span><span><span>^</span></span></span></span></span></span><span>)</span><span>⋅</span></span><span><span><span>e</span><span><span><span><span><span><span><span>P</span><span>(</span><span>s</span><span>)</span></span></span></span></span></span></span></span><span>,</span></span></span></span></span></p> <p>where <span><span>detζ\det_\zeta</span><span><span><span>det<span><span><span><span><span><span>ζ</span></span></span><span></span></span></span></span></span></span></span></span> is the zeta-regularized determinant and <span><span>P(s)P(s)</span><span><span><span>P</span><span>(</span><span>s</span><span>)</span></span></span></span> is an entire function. Through a combination of operator theory, Hadamard factorization, and asymptotic analysis, we argue that this identity is globally valid, and that the Riemann Hypothesis follows as a spectral consequence.</p> <p>The paper includes:</p> <ul> <li> <p>An explicit operator construction from first principles;</p> </li> <li> <p>Determinant identities derived from spectral zeta functions;</p> </li> <li> <p>Proof sketches connecting operator spectrum to zeta-zeros;</p> </li> <li> <p>Conditions for uniqueness and completeness of the operator;</p> </li> <li> <p>A roadmap toward a rigorous, non-circular spectral proof of the Riemann Hypothesis.</p> </li> </ul> <p>This is a formal preprint outlining a concrete and falsifiable approach. Comments and critiques are welcome.<br>For feedback, contact: <strong><a href="mailto:alex21259alex@gmail.com" rel="noopener">alex21259alex@gmail.com</a></strong></p>