The Riemann $Ξ$-function from primitive Markovian cycles I: A canonical construction
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914299608301568 |
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| author | Watson, Douglas F. |
| author_facet | Watson, Douglas F. |
| contents | Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a scaling-limit renormalized trace kernel admitting an exact theta-series representation. The construction is entirely Archimedean and uses no Euler products, primes, or arithmetic spectral input. From this limit we define a logarithmic kernel $Φ$ and prove that it lies in the Pólya frequency class $\mathrm{PF}_\infty$, yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-Pólya function $Ψ$. Independently, we introduce an Archimedean completion operator and show that, at a self-dual normalization, the completed kernel coincides with the classical theta kernel, whose Mellin transform is the Riemann $Ξ$-function. We isolate a single remaining analytic problem relating $Ψ$ to $Ξ(2\cdot)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01248 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Riemann $Ξ$-function from primitive Markovian cycles I: A canonical construction Watson, Douglas F. General Mathematics [2020]{60J27, 33E05, 42A38, 26C10, 15B48} Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a scaling-limit renormalized trace kernel admitting an exact theta-series representation. The construction is entirely Archimedean and uses no Euler products, primes, or arithmetic spectral input. From this limit we define a logarithmic kernel $Φ$ and prove that it lies in the Pólya frequency class $\mathrm{PF}_\infty$, yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-Pólya function $Ψ$. Independently, we introduce an Archimedean completion operator and show that, at a self-dual normalization, the completed kernel coincides with the classical theta kernel, whose Mellin transform is the Riemann $Ξ$-function. We isolate a single remaining analytic problem relating $Ψ$ to $Ξ(2\cdot)$. |
| title | The Riemann $Ξ$-function from primitive Markovian cycles I: A canonical construction |
| topic | General Mathematics [2020]{60J27, 33E05, 42A38, 26C10, 15B48} |
| url | https://arxiv.org/abs/2602.01248 |