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By a theorem of Bayart, $\varphi$ generates a bounded composition operator onthe Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if$\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ aDirichlet series with the following mapping properties: $\psi$ maps the righthalf-plane into the half-plane $\Real s >1/2$ if $c_0=0$ and is eitheridentically zero or maps the right half-plane into itself if $c_0$ is positive.It is shown that the $n$th approximation numbers of bounded compositionoperators on $\Hp$ are bounded below by a constant times $r^n$ for some $0<r<1$when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when$c_0$ is positive. Both results are best possible. Estimates rely on acombination of soft tools from Banach space theory ($s$-numbers, type andcotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certaininterpolation method for $\Ht$, developed in an earlier paper, using estimatesof solutions of the $\overline{\partial}$ equation. A transference principlefrom $H^p$ of the unit disc is discussed, leading to explicit examples ofcompact composition operators on $\Ho$ with approximation numbers decaying at avariety of sub-exponential rates. Finally, a new Littlewood--Paley formula isestablished, yielding a sufficient condition for a composition operator on$\Hp$ to be compact.

Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series