The Riemann zeta function ζ(s) is defined as the infinite sum
∑n=1∞n−s, which converges when Res>1. The Riemann
hypothesis asserts that the nontrivial zeros of ζ(s) lie on the line
Res=21. 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
Res<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).Comment: 8 pages, version to appear in J. Pays.