Completed Dyadic Zeta Functions with Cosine and Sine Kernels: Functional Symmetry and Zero Distributions Modeled on Riemann's ξ(s)
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2025
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| _version_ | 1866901394004377600 |
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| author | Odeh, Salaheddin |
| author_facet | Odeh, Salaheddin |
| contents | <p>This paper presents two analytically completed dyadic representations of the Riemann zeta function, constructed from trigonometric kernel decompositions over dyadic prime products. The first, denoted $\xi_{cc}(s)$, is based on non-alternating cosine kernels and exhibits even symmetry, while the second, $\xi_{cs}(s)$, employs sine kernels to encode alternating structures with odd symmetry. Both constructions are designed to emulate the reflection symmetry of Riemann’s completed zeta function $\xi(s)$ about the critical line $\Re(s) = \tfrac{1}{2}$.</p> <p>These dyadic formulations yield explicitly factorizable infinite products with built-in spectral symmetry and dyadic scaling, providing a symbolic framework for studying zeta-function behavior beyond the classical Euler product. The zero distributions of both functions form vertically aligned and symmetrically reflected sets along $\Re(s) = 0$ and $\Re(s) = 1$, offering alternative perspectives to the Riemann Hypothesis. A comparative analysis reveals deep structural dualities and motivates future research into modular, spectral, and symbolic generalizations of analytic number theory.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15484570 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Completed Dyadic Zeta Functions with Cosine and Sine Kernels: Functional Symmetry and Zero Distributions Modeled on Riemann's ξ(s) Odeh, Salaheddin Riemann zeta function dyadic prime product functional equation spectral symmetry cosine kernel sine kernel Möbius regularization alternating zeta structures zero distribution symbolic analytic continuation <p>This paper presents two analytically completed dyadic representations of the Riemann zeta function, constructed from trigonometric kernel decompositions over dyadic prime products. The first, denoted $\xi_{cc}(s)$, is based on non-alternating cosine kernels and exhibits even symmetry, while the second, $\xi_{cs}(s)$, employs sine kernels to encode alternating structures with odd symmetry. Both constructions are designed to emulate the reflection symmetry of Riemann’s completed zeta function $\xi(s)$ about the critical line $\Re(s) = \tfrac{1}{2}$.</p> <p>These dyadic formulations yield explicitly factorizable infinite products with built-in spectral symmetry and dyadic scaling, providing a symbolic framework for studying zeta-function behavior beyond the classical Euler product. The zero distributions of both functions form vertically aligned and symmetrically reflected sets along $\Re(s) = 0$ and $\Re(s) = 1$, offering alternative perspectives to the Riemann Hypothesis. A comparative analysis reveals deep structural dualities and motivates future research into modular, spectral, and symbolic generalizations of analytic number theory.</p> |
| title | Completed Dyadic Zeta Functions with Cosine and Sine Kernels: Functional Symmetry and Zero Distributions Modeled on Riemann's ξ(s) |
| topic | Riemann zeta function dyadic prime product functional equation spectral symmetry cosine kernel sine kernel Möbius regularization alternating zeta structures zero distribution symbolic analytic continuation |
| url | https://doi.org/10.5281/zenodo.15484570 |