Operator-valued zeta functions and Fourier analysis

The Riemann zeta function ζ(s) is defined as the infinite sum sum_n=1^infty n^-s, which converges when {rm Re} s>1. The Riemann hypothesis asserts that the nontrivial zeros of ζ(s) lie on the line {rm Re} s= frac{1}{2}. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex s for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region {rm Re} s<1 by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by ζ(s).