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

2022/06/06 by Gallardo-Gutiérrez, Eva, Seco, Daniel · 1 citation
#11M26 #30H20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.02654

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 Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces D_α when α∈ (-3,-2) give conditions so that the half-plane \s ∈ ℂ: \Re (s) > -(α+1)/(2)\ is also zero-free 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.

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• Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects

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