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

TL;DR

This paper links zero-free regions of the Riemann zeta function to approximation problems in weighted Dirichlet spaces, extending known results and providing new formulations of the Prime Number Theorem.

Contribution

It generalizes zero-free region results for the zeta function to a broader class of weighted Dirichlet spaces and related function spaces.

Findings

Zero-free regions are characterized for spaces D_α with α in (-3,-2).

Approximation problems in these spaces relate to the zeta function's zeros.

A new formulation of the Prime Number Theorem is derived for p=1, α=-2.

Abstract

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 $\{s \in \mathbb{C}: \Re (s) > -\frac{\alpha+1}{2}\}$ 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.