Operator-valued zeta functions and Fourier analysis

The Riemann zeta function Ϛ(s) is defined as the infinite sum ∑∞n=1n-s, which converges when Re s ˃ 1. The Riemann hypothesis asserts that the nontrivial zeros of Ϛ(s) lie on the line Re s = ½ . 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 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 Ϛ(s)