Distribution of primes and approximation on weighted Dirichlet spaces

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Gallardo Gutiérrez, Eva A. and Seco, Daniel (2022) Distribution of primes and approximation on weighted Dirichlet spaces. (Unpublished)

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Abstract

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.


Item Type:Article
Uncontrolled Keywords:Riemann zeta function; weighted Dirichlet spaces; cyclic vectors
Subjects:Sciences > Mathematics
Sciences > Mathematics > Mathematical analysis
ID Code:73278
Deposited On:30 Jun 2022 10:49
Last Modified:30 Jun 2022 11:21

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