2025/11/10 by Ken Nagai, Nagai, Ken
Computer Science · Mathematics · #11B68(Primary) #33E17 #34B24 #47B10 #60B20(Secondary) #FOS: Mathematics #General Mathematics (math.GM) #Quantum Information and Cryptography #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2511.07495
openalex publication_date 2025/11/10 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28
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.