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arXiv · 1412.5947

The Dirichlet-Bohr radius

Daniel Carando·Andreas Defant·Domingo García·Manuel Maestre·Pablo Sevilla-Peris
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Abstract

Denote by $Ω(n)$ 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 $$ \sum_{n \leq x} |a_n| r^{Ω(n)} \leq \sup_{t\in \mathbb{R}} \big|\sum_{n \leq x} a_n n^{-it}\big|\,. $$ 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 to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.

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Daniel Carando, Andreas Defant, Domingo García, Manuel Maestre, Pablo Sevilla-Peris. 2014-12-16. The Dirichlet-Bohr radius. https://doi.org/10.4064/aa171-1-3

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