Zero-Free Regions of the Riemann Zeta Function and Approximation in Weighted Dirichlet Spaces
Articles
Overview
published in
publication date
- January 2025
start page
- 1
end page
- 27
issue
- 38
volume
- 19
Digital Object Identifier (DOI)
International Standard Serial Number (ISSN)
- 1661-8254
Electronic International Standard Serial Number (EISSN)
- 1661-8262
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 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 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.
Classification
subjects
- Mathematics
keywords
- riemann zeta function; weighted dirichlet spaces; cyclic vectors