The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series

Given a frequency $\lambda =({\lambda }_n)$, we consider the Hardy spaces ${\mathcal{H}}_p^{\lambda }$ of $\lambda$-Dirichlet series $D={\sum }_n{a}_n{e}^{-{\lambda }_{n}s}$ and study the asymptotic behavior of the upper and lower democracy functions of its canonical basis $\mathcal{B}=\{e^{-{\lambda }_{n}s}\}$. For the ordinary case, $\mathcal{B}=\{n^{-s}\}$, we give the correct asymptotic behavior of all such functions, while in the general case we give sharp lower and upper bounds for all possible behaviors. Moreover, for $p>2$ we present examples showing that any intermediate behavior (between the extreme bounds) can occur. We also study how different properties of the frequency $\lambda$ lead to particular behaviors of the corresponding fundamental functions. Finally, we apply our results to analyze greedy-type properties of $\mathcal{B}=\{e^{-{\lambda }_{n}s}\}$ for some particular $\lambda$’s.