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Abstract. For α ∈ R, let Dα denote the scale of Hilbert spaces consisting of Dirichlet series f (s) =The Gordon-Hedenmalm Theorem on composition operators for H 2 = D 0 is extended to the Bergman case α > 0. These composition operators are generated by functions of the form Φ(s) = c 0 s+ϕ(s), where c 0 is a nonnegative integer and ϕ(s) is a Dirichlet series with certain convergence and mapping properties. For the operators with c 0 = 0 a new phenomenon is discovered: If 0 < α < 1, the space Dα is mapped by the composition operator into a smaller space in the same scale. When α > 1, the space Dα is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet-Bergman spaces A p for 1 ≤ p < ∞ are obtained, in addition to new partial results for composition operators on the DirichletHardy spaces H p when p is an odd integer.

ABSTRACT. For a Dirichlet series symbol g (s) = n≥1 b n n −s , the associated Volterra operator T g acting on a Dirichlet series f (s) = n≥1 a n n −s is defined by the integral f → − +∞ s f (w)g ′ (w) d w. We show that T g is a bounded operator on the Hardy space H p of Dirichlet series with 0 < p < ∞ if and only if the symbol g satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of BMOA(D). A further analogy with classical BMO is that exp(c|g |) is integrable (on the infinite polytorus) for some c > 0 whenever T g is bounded.In particular, such g belong to H p for every p < ∞. We relate the boundedness of T g to several other BMO type spaces: BMOA in halfplanes, the dual of H 1 , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of T g on reproducing kernels for appropriate sequences of subspaces of H 2 . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g .

We study H p spaces of Dirichlet series, called H p , for the range 0 < p < ∞. We begin by showing that two natural ways to define H p coincide. We then proceed to study some linear space properties of H p . More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy-Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between H p and H 4/p , contrasting the usual L p duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of H p and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator ∞ n=1 a n n −s → N n=1 a n n −s on H p with 0 < p ≤ 1, supplementing a classical result of Helson for the range 1 < p < ∞. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges 1 ≤ p ≤ ∞ and 0 < p < 1.