Zero-free half-planes of the ζ-function via spaces of analytic functions

In this article we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function ζ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted ${\ell }^2$ spaces and the classical Hardy spaces $H^p$ ($0<p\le 2$). As a consequence precise conditions are obtained for the existence of zero-free half planes for the ζ-function.