From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, {\zeta}-Regularization, and the Bridge Between Discrete and Continuous Spectra

TL;DR

This paper develops a unified framework linking Bernoulli numbers, zeta-regularization, and Fredholm determinants, revealing deep connections between discrete combinatorics and continuous spectral theory, with implications for random matrix models and special functions.

Contribution

It introduces a novel analytic approach connecting Bernoulli numbers, spectral traces, and Fredholm determinants, bridging discrete and continuous spectral phenomena.

Findings

Fredholm determinants interpolate between finite-rank projectors and sine-kernels.

In the continuum limit, the determinant becomes a Painleve-V tau-function.

Bernoulli coefficients and zeta-constants describe spectral asymptotics.

Abstract

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.