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We investigate the compactness of composition operators on the Hardy space ofDirichlet series induced by a map $\varphi(s)=c_0s+\varphi_0(s)$, where$\varphi_0$ is a Dirichlet polynomial. Our results depend heavily on thecharacteristic $c_0$ of $\varphi$ and, when $c_0=0$, on both the degree of$\varphi_0$ and its local behaviour near a boundary point. We also study theapproximation numbers for some of these operators. Our methods involvegeometric estimates of Carleson measures and tools from differential geometry.

Compact composition operators with non-linear symbols on the $H^2$ space of Dirichlet series