  Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset
\mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all
Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in
\mathbb{C}$. The main aim is to prove asymptotically correct estimates for the
projection constant $\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega)
\big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(\omega)$
equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic
analysis on $\omega$-Dirichlet groups, we prove the formula $
\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big) = \lim_{T \to
\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 $J$. To see an example, combining with a recent deep result of
Harper from probabilistic analytic number theory, we for the space
$\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet
polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the
asymptotically correct order $
\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big)
\sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $
