%0 Journal Article
%A Gallardo Gutiérrez, Eva A.
%A Seco, Daniel
%T Distribution of primes and approximation on weighted Dirichlet spaces
%D 2022
%U https://hdl.handle.net/20.500.14352/71714
%X 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 B ́aez-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 : R(s) > − α+12} is also zero-free for ζ. Moreover, we extend such results to a large family of weighted spaces of analytic functions lp α. As a particular instance, in the limit case p = 1 and α = −2, we provide a new proof of the Prime Number Theorem.
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