Determining the Indeterminate: On the Evaluation of Integrals that connect Riemann's, Hurwitz' and Dirichlet's Zeta, Eta and Beta functions (2107.12559v7)

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 and their alternating equivalents. Interesting special cases involving improper integrals containing $\zeta(\sigma+it)$ and a minimum of other functions in the integrand are identified. Many of these integrals do not appear in the literature and can be 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, but probably is not. 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 studies this singularity from several angles in an attempt to resolve the associated ambiguities. This is done by establishing a self-consistent way to remove the singularity and thereby evaluate members of the family. These new closed-form identities provide insight into the value of integrals of general interest. Some implications are proposed.