# Infinite Products and Zeta-Regularization

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## Abstract

Our purpose here is to review some symmetric-function techniques suitable for certain *divergent* sequences \(\{x_{k}\} k = 1,2,\ldots,\) i.e., infinite (real or complex) sequences with \(\left|x_{k}\right| \uparrow \infty\). (The *x*_{ k } are counted with multiplicities if any.) We specifically wish to control their symmetric functions of the Zeta and Delta types as sketched in Sect. 1.1. We will elaborate on a scheme initiated for positive eigenvalue spectra in [104], where Theta-type functions provided a natural and most convenient base. For sequences such as the Riemann zeros, however, Theta-type functions exhibit less accessible and rather intricate properties, whereas Delta-type functions are openly present (as the “trivial” Gamma factors) and thus provide a privileged gateway; at the same time, a setting which requires positive sequences is inadequate. We must then adapt [104] to a broader perspective better adjusted to the idiosyncrasies of sequences like the Riemann zeros, and favoring conditions placed on Delta-type functions. The latter option, however, lengthens some intermediate calculations, so we emphasize that these are totally elementary (nineteenth-century mathematics!), all the more that a parameter \(\mu_{0}\), the order of the Delta function, remains low; and only \(\mu_{0} = 1 \quad {\rm and} \quad \frac{1}{2}\) will serve for the Riemann zeros. Our goal here is to output a toolbox of basic special-value formulae that are general enough and systematic yet economical, being tailored to our current final needs. Alternatively, we refer to [32,53,60,61,90] for very powerful and general, but more elaborate, frameworks. For greater convenience, Sect. 2.6 groups the main practical results to be exported for later use. Thus, the bulk of this chapter may be skipped on first reading. (Inversely, it can serve as a tutorial for zeta-regularization alone.)

## Keywords

Entire Function Zeta Function Integer Point Meromorphic Continuation Eligibility Condition## Preview

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