We study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman-type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz-type lemma for Dirichlet series.

We study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose, we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman-type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz-type lemma for Dirichlet series.

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The following extension of Bohr’s theorem is established: If a somewhere convergent Dirichlet series f f has an analytic continuation to the half-plane C θ = { s = σ + i t : σ > θ } \mathbb {C}_\theta = \{s = \sigma +it\,:\, \sigma >\theta \} that maps C θ \mathbb {C}_\theta to C ∖ { α , β } \mathbb {C} \setminus \{\alpha ,\beta \} for complex numbers α ≠ β \alpha \neq \beta , then f f converges uniformly in C θ + ε \mathbb {C}_{\theta +\varepsilon } for any ε > 0 \varepsilon >0 . The extension is optimal in the sense that the assertion no longer holds should C ∖ { α , β } \mathbb {C}\setminus \{\alpha ,\beta \} be replaced with C ∖ { α } \mathbb {C}\setminus \{\alpha \} .