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 Dα when α ∈ (−3, −2) give conditions so that the half-plane {s ∈ C : (s) > −α+1 /2 } is also zerofree for ζ . Moreover, we extend such results to a large family of weighted spaces of analytic functions ℓ 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.

Zero-Free Regions of the Riemann Zeta Function and Approximation in Weighted Dirichlet Spaces Articles