Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces
We study zero-free regions of the Riemann zeta function $ζ$ 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 $D_α$ when $α\in (-3,-2)$ give conditions so that the half-plane $\{s \in \mathbb{C}: \Re (s) > -\frac{α+1}{2}\}$ is also zero-free for $ζ$. Moreover, we extend such results to a large family of weighted spaces of analytic functions $\ell^p_α$. As a particular instance, in the limit case $p=1$ and $α=-2$, we provide a new equivalent formulation of the Prime Number Theorem.
math.NT↗