The Dirichlet-Bohr radius

Denote by $\mathrm{\Omega }(n)$ 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 ${\sum }_{n\le x}{a}_n{n}^{-s}$ we have $$\sum _{n\le x}\mathrm{\mid }{a}_n\mathrm{\mid }{r}^{\mathrm{\Omega }(n)}\le \underset{t\in \mathbb{R}}{\sup }\mid \sum _{n\le x}{a}_n{n}^{-it}\mid \mathrm{.}$$ We prove that the asymptotically correct order of $L(x)$ is $(\log x{)}^{1\mathrm{/}4}{x}^{-1\mathrm{/}8}$. 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.