Determining the Indeterminate: On the Evaluation of Integrals that connect Riemann's, Hurwitz' and Dirichlet's Zeta, Eta and Beta functions

TL;DR

This paper explores integrals involving Riemann, Hurwitz, Dirichlet zeta, eta, and beta functions, revealing new relationships, addressing indeterminism due to essential singularities, and proposing methods to evaluate these complex integrals.

Contribution

It introduces a novel approach to handle indeterminate integrals connected to zeta functions by analyzing essential singularities and establishing a self-consistent evaluation method.

Findings

Identified new integral relationships involving zeta and eta functions.

Discovered an essential singularity causing indeterminism in certain integrals.

Proposed a method to evaluate integrals containing zeta functions despite singularities.

Abstract

By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their alternating equivalents $\eta(s)$ and $\eta(s,a)$. In particular, special cases involving improper integrals containing $\zeta(\sigma+it)$ and few other functions in the integrand are identified. Many of these integrals that do not appear in the literature, can be, and were, verified numerically. In one limit, the use of analytic continuation generates a family of improper integrals containing only the real and imaginary parts of $\zeta(\sigma+it)$ with and without simple trigonometric factors; the associated closed form contains an (unclassified) entity that has many of the attributes of an essential singularity. Consequently, this means that the associated integrals are indeterminate (i.e. non-single valued), so a new symbol is introduced to label the indeterminism. Much of this paper examines this singularity from several angles in order to resolve the associated ambiguities, before eventually showing how it blends into the classical study of functions of real and complex variables in an unusual manner. This is done by establishing a self-consistent way to remove the singularity and thereby evaluate new members of a family of integrals of general interest that contain $\zeta(s,a)$ and $\eta(s,a)$. Some implications are proposed.