Operator-valued zeta functions and Fourier analysis

The Riemann zeta function $\zeta (s)$ is defined as the infinite sum ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{-s}$, which converges when $\mathrm{R}\mathrm{e}s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta (s)$ lie on the line $\mathrm{R}\mathrm{e}s=\frac{1}{2}$. 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 $\mathrm{R}\mathrm{e}s<1$ 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 $\zeta (s)$.