标题： $ζ$-函数通过围线积分和通用求和规则 显示英文标题

标题： $ζ$-functions via contour integrals and universal sum rules

摘要： 本工作开发了一个分析框架，用于研究与复数序列相关的$\zeta$函数。我们证明了当研究与自伴微分算子相关的谱$\zeta$函数时，常用的围道积分表示可以远远超出其传统设置。与利用$\theta$函数积分的表示不同，我们的方法适用于任意复数序列，只需极少的假设。这导致了一组普遍恒等式，包括求和规则和亚纯性质，这些恒等式在广泛的$\zeta$函数类中成立。此外，我们讨论了与正则化（修改）弗雷德霍姆行列式的联系，这些行列式属于$p$-Schatten--von Neumann 类算子。我们通过计算各种复数序列的$\zeta$函数的特殊值和留数来展示该表示的灵活性，特别是艾里函数、抛物柱函数和合流超几何函数的零点。此外，我们在研究艾里$\zeta$函数时采用了自适应 Antoulas--Anderson (AAA) 算法进行有理插值。 显示英文摘要

摘要： 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.