From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, ζ-Regularization, and the Bridge Between Discrete and Continuous Spectra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917071500083200 |
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| author | Nagai, Ken |
| author_facet | Nagai, Ken |
| contents | We construct a unified analytic framework connecting Bernoulli numbers, zeta-regularization, and Fredholm determinants associated with trigonometric selector kernels. Starting from the Bernoulli-Stirling algebra, Euler-Maclaurin corrections are reinterpreted as spectral traces of compact operators. This bridge transforms discrete combinatorial data into continuous spectral quantities, showing that their determinants interpolate between finite-rank projectors and the sine-kernel of random-matrix theory. In the continuum limit the Fredholm determinant becomes a Painleve-V~tau-function, revealing a hierarchy in which Bernoulli coefficients and zeta-constants jointly describe the local-global asymptotics of analytic regularization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, ζ-Regularization, and the Bridge Between Discrete and Continuous Spectra Nagai, Ken General Mathematics 11B68(Primary), 47B10, 34B24, 33E17, 60B20(Secondary) We construct a unified analytic framework connecting Bernoulli numbers, zeta-regularization, and Fredholm determinants associated with trigonometric selector kernels. Starting from the Bernoulli-Stirling algebra, Euler-Maclaurin corrections are reinterpreted as spectral traces of compact operators. This bridge transforms discrete combinatorial data into continuous spectral quantities, showing that their determinants interpolate between finite-rank projectors and the sine-kernel of random-matrix theory. In the continuum limit the Fredholm determinant becomes a Painleve-V~tau-function, revealing a hierarchy in which Bernoulli coefficients and zeta-constants jointly describe the local-global asymptotics of analytic regularization. |
| title | From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, ζ-Regularization, and the Bridge Between Discrete and Continuous Spectra |
| topic | General Mathematics 11B68(Primary), 47B10, 34B24, 33E17, 60B20(Secondary) |
| url | https://arxiv.org/abs/2511.07495 |