Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces

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 B'aez-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 $\{s\in \mathbb{C}:\mathrm{\Re }(s)>-\frac{\alpha +1}{2}\}$ is also zero-free for $\zeta$. Moreover, we extend such results to a large family of weighted spaces of analytic functions ${\mathrm{\ell }}_{\alpha }^p$. As a particular instance, in the limit case $p=1$ and $\alpha =-2$, we provide a new equivalent formulation of the Prime Number Theorem.