Zero-Free Regions of the Riemann Zeta Function and Approximation in Weighted Dirichlet Spaces Articles uri icon

authors

  • Gallardo-Gutierrez, Eva A.
  • SECO FORSNACKE, DANIEL

publication date

  • January 2025

start page

  • 1

end page

  • 27

issue

  • 38

volume

  • 19

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.

subjects

  • Mathematics

keywords

  • riemann zeta function; weighted dirichlet spaces; cyclic vectors