The Dirichlet–Bohr radius
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
