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We give necessary and sufficient conditions for a composition operator withDirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardyspace $\mathcal{H}^2$ of Dirichlet series. For $p\geq 2$, these conditions leadto a characterization for the subclass of symbols with bounded imaginary parts.Finally, we establish a comparison-type principle for composition operators.Applying our techniques in conjunction with classical geometric function theorymethods, we prove the analogue of the polygonal compactness theorem for$\mathcal{H}^2$ and we give examples of bounded composition operators withDirichlet series symbols on $\mathcal{H}^p,\,p>0$.

Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle