$\zeta$-functions via contour integrals and universal sum rules

This work develops an analytic framework for the study of the $\zeta$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $\zeta$-functions associated with self-adjoint differential operators, can be extended far beyond its traditional setting. In contrast to representations utilizing integrals of $\theta$-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 $\zeta$-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 $\zeta$-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 $\zeta$-function.