The Dirichlet-Bohr radius

TL;DR

This paper determines the asymptotic behavior of the Dirichlet-Bohr radius, linking it to classical Bohr radii for multivariable holomorphic functions and establishing a systematic translation between these concepts.

Contribution

It establishes the asymptotic order of the Dirichlet-Bohr radius and introduces a framework connecting it with classical Bohr radii for holomorphic functions.

Findings

Asymptotic order of L(x) is (log x)^{1/4} x^{-1/8}

Links Dirichlet-Bohr radii with classical Bohr radii

Provides a systematic translation framework between the two concepts

Abstract

Denote by $\Omega(n)$ 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 $$ \sum_{n \leq x} |a_n| r^{\Omega(n)} \leq \sup_{t\in \mathbb{R}} \big|\sum_{n \leq x} a_n n^{-it}\big|\,. $$ 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.