2023-11-29T06:21:12Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/717142023-07-18T02:50:48Zcom_20.500.14352_14col_20.500.14352_15
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Gallardo Gutiérrez, Eva A.
author
Seco, Daniel
author
2022
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.
https://hdl.handle.net/20.500.14352/71714
Distribution of primes and approximation on weighted Dirichlet spaces