We study zero-free regions of the Riemann zeta function \(\zeta \) related to an approximation problem in the weighted Dirichlet space \(D_{-2}\) which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces \(D_{\alpha }\) when \(\alpha \in (-3,-2)\) give conditions so that the half-plane \(\{s \in \mathbb {C}: \Re (s) > -\frac{\alpha +1}{2}\}\) is also zero-free for \(\zeta \) . Moreover, we extend such results to a large family of weighted spaces of analytic functions \(\ell ^p_{\alpha }\) . As a particular instance, in the limit case \(p=1\) and \(\alpha =-2\) , we provide a new equivalent formulation of the Prime Number Theorem.
