The Dirichlet-Bohr radius

Daniel Carando, et al. "The Dirichlet-Bohr radius." Acta Arithmetica 171.1 (2015): 23-37. <http://eudml.org/doc/279056>.

@article{DanielCarando2015,
abstract = {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 $∑_\{n ≤ x\} a_n n^\{-s\}$ we have $∑_\{n ≤ x\} |a_n| r^\{Ω(n)\} ≤ sup_\{t∈ ℝ\} |∑_\{n ≤ x\} a_n n^\{-it\}|$. We prove that the asymptotically correct order of L(x) is $(log x)^\{1/4\} x^\{-1/8\}$. 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.},
author = {Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris},
journal = {Acta Arithmetica},
language = {eng},
number = {1},
pages = {23-37},
title = {The Dirichlet-Bohr radius},
url = {http://eudml.org/doc/279056},
volume = {171},
year = {2015},
}

TY - JOUR
AU - Daniel Carando
AU - Andreas Defant
AU - Domingo A. Garcí
AU - Manuel Maestre
AU - Pablo Sevilla-Peris
TI - The Dirichlet-Bohr radius
JO - Acta Arithmetica
PY - 2015
VL - 171
IS - 1
SP - 23
EP - 37
AB - 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 $∑_{n ≤ x} a_n n^{-s}$ we have $∑_{n ≤ x} |a_n| r^{Ω(n)} ≤ sup_{t∈ ℝ} |∑_{n ≤ x} a_n n^{-it}|$. We prove that the asymptotically correct order of L(x) is $(log x)^{1/4} x^{-1/8}$. 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.
LA - eng
UR - http://eudml.org/doc/279056
ER -