Abstract
It is remarkable that the quantum zeta function, defined as a sum over energy eigenvalues E:
admits of exact evaluation in some situations for which not a single E be known. Herein we show how to evaluate instances of Z(s), and of an associated parity zeta function Y(s), for various quantum systems. For some systems both Z(n), Y(n) can be evaluated for infinitely many integers n. Such Z,Y values can be used, for example, to effect sharp numerical estimates of a system's ground energy. The difficult problem of evaluating the analytic continuation Z(s) for arbitrary complex s is discussed within the contexts of perturbation expansions, path integration, and quantum chaos.