The Bessel zeta function

TL;DR

This paper investigates two representations of the Bessel zeta function, deriving new series representations and connecting them to the Riemann zeta function through analytic continuation, providing insights into their structure and properties.

Contribution

It introduces a fully evaluated integral representation of the Bessel zeta function and explores its relation to the Riemann zeta function via analytic continuation.

Findings

Derived two similar series representations of the Bessel zeta function.

Validated the representations against known values, slopes, and pole structures.

Connected the Bessel zeta function to the Riemann zeta function through order limit analysis.

Abstract

Two representations of the Bessel zeta function are investigated. An incomplete representation is constructed using contour integration and an integral representation due to Hawkins is fully evaluated (analytically continued) to produce two infinite series. This new representation, evaluated at integer values of the argument, produces results that are consistent with known results (values, slope, and pole structure). Not surprisingly, the two representations studied are found to have similar coefficients but a slightly different functional form. A representation of the Riemann zeta function is obtained by allowing the order of the Bessel function to go to 1/2.