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The Dirichlet–Bohr radius

Volume 171 / 2015

Daniel Carando, Andreas Defant, Domingo a Garcí, Manuel Maestre, Pablo Sevilla-Peris Acta Arithmetica 171 (2015), 23-37 MSC: 11M41, 30B50, 11M36. DOI: 10.4064/aa171-1-3

Abstract

Denote by $\varOmega(n)$ the number of prime divisors of $n \in \mathbb{N}$ (counted with multiplicities). For $x\in \mathbb{N}$ define the Dirichlet–Bohr radius $L(x)$ to be the best $r>0$ such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_n n^{-s}$ we have $$ \sum_{n \leq x} |a_n| r^{\varOmega(n)} \leq \sup_{t\in \mathbb{R}}\, \Bigl|\sum_{n \leq x} a_n n^{-it}\Bigr|. $$ 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.

Authors

  • Daniel CarandoDepartamento de Matem\'atica
    Facultad de Ciencias Exactas y Naturales
    Universidad de Buenos Aires
    Pab I, Ciudad Universitaria
    1428, Buenos Aires, Argentina
    and
    IMAS – CONICET
    e-mail
  • Andreas DefantInstitut für Mathematik
    Universität Oldenburg
    D-26111 Oldenburg, Germany
    e-mail
  • Domingo a GarcíDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Manuel MaestreDepartamento de Análisis Matemáatico
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Pablo Sevilla-PerisInstituto Universitario de Matemática Pura y Aplicada
    Universitat Politécnica de Valéncia
    Valencia, Spain
    e-mail

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