Mellin Transform Techniques for Zeta-Function Resummations

TL;DR

This paper introduces Mellin transform techniques for zeta-function resummations, providing a new proof and extension of the zeta regularization theorem applicable to elliptic operators, with practical applications in string theory partition functions.

Contribution

It presents an elegant Mellin transform-based proof of the zeta regularization theorem, extending its applicability to elliptic operators with unknown spectra, and offers analytical formulas and numerical estimates.

Findings

New proof of zeta regularization theorem using Mellin transforms

Extension to elliptic operators with unknown spectra

Application to string theory partition function summation

Abstract

Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the result is naturally split into the same three contributions of very different nature, i.e. the series of Riemann zeta functions and the power and negative exponentially behaved functions, respectively, well known from the original proof. The new theorem deals equally well with elliptic differential operators whose spectrum is not explicitly known. Rigorous results on the asymptoticity of the outcoming series are given, together with some specific examples. Exact analytical formulas, simplifying approximations and numerical estimates for the last of the three contributions (the most difficult to handle) are obtained. As an application of the method, the summation of the series which appear in the analytic computation (for different ranges of temperature) of the partition function of the string ---basic in order to ascertain if QCD is some limit of a string theory--- is PERFORMED.