  In this paper, we completely give the solution of the problem of
Littlewood-type randomization in the Hardy and Bergman spaces of Dirichlet
series. The Littlewood-type theorem for Bergman spaces of Dirichlet series is
very different from the corresponding version for Hardy spaces of Dirichlet
series; but also exhibits various pathological phenomena compared with the
setting of analytic Bergman spaces over the unit disk, due to the fact that
Dirichlet series behave as power series of infinitely many variables. A
description for the inclusion between some mixed norm spaces of Dirichlet
series plays an essential role in our investigation. Finally, as another
application of the inclusion, we completely characterize the superposition
operators between Bergman spaces of Dirichlet series.
