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

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