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.
