Zero-free half-planes of the \zeta -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 $\zeta$ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted $\ell^2$ spaces and classical Hardy spaces $H^p$ ($0<p\leq2$). As a consequence precise conditions are obtained for the existence of zero-free half planes for the $\zeta$-function.