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 $\mathcal B=\{ e^{ -\lambda_ns}\}$. 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 $\mathcal B=\{ e^{ -\lambda_ns}\}$ for some particular $\lambda$'s.
Categories: math.FA