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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(Unpublished)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 |
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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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