From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, ζ-Regularization, and the Bridge Between Discrete and Continuous Spectra

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.

Figures (10)

• Figure 1: Schematic view of the Euler--Maclaurin bridge: rectangles (discrete sums) versus area (integral). Bernoulli numbers measure their difference.
• Figure 2: Triangular correspondence among the three regularizations: local (Bernoulli), global ($\zeta$), and operator (Fredholm).
• Figure 3: From discrete arrays to kernels: the passage from combinatorics to operator theory.
• Figure 4: Matrix $\rightarrow$ kernel correspondence: as dimension grows, discrete indices $(j,k)$ become continuous variables $(x,y)$.
• Figure 5: From finite determinants to Fredholm determinants: matrix $\det$ extends analytically to operators $\det\nolimits_{\!F}\!\left(\cdot\right)$.

Theorems & Definitions (17)

• Remark 1.1: Bridge in one sentence
• Example 2.1: Quick check
• Remark 2.2: From array to operator
• Remark 2.3: Takeaway
• Example 3.1: A canonical kernel
• Definition 3.2: Fredholm determinant
• Remark 3.3: Analogy to the zeta determinant
• Example 3.4: Rank–one kernel
• Example 3.5: Orthogonal projector
• Remark 3.6: Key picture