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...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autores principales: Carando, D., Defant, A., García, D., Maestre, M., Sevilla-Peris, P.
Formato: JOUR
Materias:
Bohr radius
Dirichlet series
Holomorphic functions
Acceso en línea: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 todo:paper_00651036_v171_n1_p23_Carando
record_format dspace
spelling todo:paper_00651036_v171_n1_p23_Carando2023-10-03T14:52:45Z The Dirichlet-Bohr radius Carando, D. Defant, A. García, D. Maestre, M. Sevilla-Peris, P. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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, D.
Defant, A.
García, D.
Maestre, M.
Sevilla-Peris, P.
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.
format JOUR
author Carando, D.
Defant, A.
García, D.
Maestre, M.
Sevilla-Peris, P.
author_facet Carando, D.
Defant, A.
García, D.
Maestre, M.
Sevilla-Peris, P.
author_sort Carando, D.
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
url http://hdl.handle.net/20.500.12110/paper_00651036_v171_n1_p23_Carando
work_keys_str_mv AT carandod thedirichletbohrradius
AT defanta thedirichletbohrradius
AT garciad thedirichletbohrradius
AT maestrem thedirichletbohrradius
AT sevillaperisp thedirichletbohrradius
AT carandod dirichletbohrradius
AT defanta dirichletbohrradius
AT garciad dirichletbohrradius
AT maestrem dirichletbohrradius
AT sevillaperisp dirichletbohrradius
_version_ 1782024392658649088

Ejemplares similares