The Bessel zeta function

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.