Operator-valued zeta functions and Fourier analysis

Abstract

The Riemann zeta function (s) is defined as the infinite sum n =1 n?s, which converges when Re s > 1. The Riemann hypothesis asserts that the nontrivial zeros of (s) lie on the line Re s = 12 . 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 (s).