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In this article, we introduce a general approach for deriving zero-freehalf-planes for the Riemann zeta function $\zeta$ by identifying topologicalvector spaces of analytic functions with specific properties. This approach isapplied to weighted $\ell^2$ spaces and classical Hardy spaces $ H^p $ ($0<p\leq2 $). As a consequence precise conditions are obtained for the existenceof zero-free half planes for the $\zeta$-function.

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