The Dirichlet-Bohr radius

Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x ∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial Σn≤xann-s we have ∑n ≤ x |an| rΩ(n) ≤ supt ∈ ℝ | ∑n ≤ x ann-it|. We prove that the asymptoti...

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Autor principal: Carando, Daniel German
Publicado: 2015
Materias:
Bohr radius
Dirichlet series
Holomorphic functions
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00651036_v171_n1_p23_Carando
http://hdl.handle.net/20.500.12110/paper_00651036_v171_n1_p23_Carando
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00651036_v171_n1_p23_Carando
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spelling paper:paper_00651036_v171_n1_p23_Carando2023-06-08T15:05:53Z The Dirichlet-Bohr radius Carando, Daniel German Bohr radius Dirichlet series Holomorphic functions Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x ∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial Σn≤xann-s we have ∑n ≤ x |an| rΩ(n) ≤ supt ∈ ℝ | ∑n ≤ x ann-it|. We prove that the asymptotically correct order of L(x) is (log x)1/4x-1/8. Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa. Copyright © 2007-2014 by IMPAN. All rights reserved. Fil:Carando, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2015 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00651036_v171_n1_p23_Carando http://hdl.handle.net/20.500.12110/paper_00651036_v171_n1_p23_Carando
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Bohr radius
Dirichlet series
Holomorphic functions
spellingShingle Bohr radius
Dirichlet series
Holomorphic functions
Carando, Daniel German
The Dirichlet-Bohr radius
topic_facet Bohr radius
Dirichlet series
Holomorphic functions
description Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x ∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial Σn≤xann-s we have ∑n ≤ x |an| rΩ(n) ≤ supt ∈ ℝ | ∑n ≤ x ann-it|. We prove that the asymptotically correct order of L(x) is (log x)1/4x-1/8. Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa. Copyright © 2007-2014 by IMPAN. All rights reserved.
author Carando, Daniel German
author_facet Carando, Daniel German
author_sort Carando, Daniel German
title The Dirichlet-Bohr radius
title_short The Dirichlet-Bohr radius
title_full The Dirichlet-Bohr radius
title_fullStr The Dirichlet-Bohr radius
title_full_unstemmed The Dirichlet-Bohr radius
title_sort dirichlet-bohr radius
publishDate 2015
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00651036_v171_n1_p23_Carando
http://hdl.handle.net/20.500.12110/paper_00651036_v171_n1_p23_Carando
work_keys_str_mv AT carandodanielgerman thedirichletbohrradius
AT carandodanielgerman dirichletbohrradius
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