The Dirichlet-Bohr radius

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.