The Dirichlet–Bohr radius

Daniel Carando ¹   ; Andreas Defant ²   ; Domingo a Garcí ³   ; Manuel Maestre ⁴   ; Pablo Sevilla-Peris ⁵

¹ Departamento de Matem\'atica Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Pab I, Ciudad Universitaria 1428, Buenos Aires, Argentina and IMAS – CONICET
² Institut für Mathematik Universität Oldenburg D-26111 Oldenburg, Germany
³ Departamento de Análisis Matemático Universidad de Valencia Doctor Moliner 50 46100 Burjasot (Valencia), Spain
⁴ Departamento de Análisis Matemáatico Universidad de Valencia Doctor Moliner 50 46100 Burjasot (Valencia), Spain
⁵ Instituto Universitario de Matemática Pura y Aplicada Universitat Politécnica de Valéncia Valencia, Spain

Acta Arithmetica, Tome 171 (2015) no. 1, pp. 23-37

Résumé

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.

Détail

DOI : 10.4064/aa171-1-3

Mots-clés : denote varomega number prime divisors mathbb counted multiplicities mathbb define dirichlet bohr radius best every finite dirichlet polynomial sum leq s have sum leq varomega leq sup mathbb bigl sum leq it bigr prove asymptotically correct order log following bohrs vision proof links estimation classical bohr radii holomorphic functions several variables moreover suggest general setting which allows translating various results bohr radii systematic results dirichlet bohr radii vice versa