Given a frequency sequence \(\omega =(\omega _n)\) and a finite subset \(J \subset {\mathbb {N}}\) , we study the space \({\mathscr {H}}_{\infty }^{J}(\omega )\) of all Dirichlet polynomials \(D(s):= \sum \nolimits _{n \in J} a_n e^{-\omega _n s}, \, s \in {\mathbb {C}}\) . The main aim is to prove asymptotically correct estimates for the projection constant \(\varvec{\lambda }\big ({\mathscr {H}}_\infty ^{J}(\omega ) \big )\) of the finite dimensional Banach space \({\mathscr {H}}_\infty ^{J}(\omega )\) equipped with the norm \(\Vert D\Vert = \sup _{\text {Re}\,s>0} |D(s)|\) . Based on harmonic analysis on \(\omega \) -Dirichlet groups, we prove the formula \( \varvec{\lambda }\big ({\mathscr {H}}_\infty ^{J}(\omega ) \big ) ~ = ~ \displaystyle \lim _{T \rightarrow \infty } \frac{1}{2T} \int _{-T}^T \Big |\sum _{n \in J} e^{-i\omega _n t}\Big |\,dt, \) and apply it to various concrete frequencies \(\omega \) and index sets J. Combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space \({\mathscr {H}}_\infty ^{\le x}\big ( (\log n)\big )\) of all ordinary Dirichlet polynomials \(D(s) = \sum _{n \le x} a_n n^{-s}\) of length x show the asymptotically correct order \( \varvec{\lambda }\big ({\mathscr {H}}_\infty ^{\le x}\big ( (\log n)\big )\big ) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. \)
