2025/08/21 by Guglielmo Fucci, Fucci, Guglielmo, Mateusz Piorkowski +3
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2508.15699
openalex publication_date 2025/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work develops an analytic framework for the study of the ζ-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral ζ-functions associated with self-adjoint differential operators, can be extended far beyond its traditional setting. In contrast to representations utilizing integrals of θ-functions, our method applies to arbitrary sequences of complex numbers with minimal assumptions. This leads to a set of universal identities, including sum rules and meromorphic properties, that hold across a broad class of ζ-functions. Additionally, we discuss the connection to regularized (modified) Fredholm determinants of p-Schatten--von Neumann class operators. We illustrate the versatility of this representation by computing special values and residues of the ζ-function for a variety of sequences of complex numbers, in particular, the zeros of Airy functions, parabolic cylinder functions, and confluent hypergeometric functions. Furthermore, we employ the adaptive Antoulas--Anderson (AAA) algorithm for rational interpolation in the study of the Airy ζ-function.