  We establish results with an arithmetic flavor that generalize the polynomial
multidimensional Szemeredi theorem and related multiple recurrence and
convergence results in ergodic theory. For instance, we show that in all these
statements we can restrict the implicit parameter to those integers that have
an even number of distinct prime factors, or satisfy any other congruence
condition. In order to obtain these refinements we study the limiting behavior
of some closely related multiple ergodic averages with weights given by
appropriately chosen multiplicative functions. These averages are then analysed
using a recent structural result for bounded multiplicative functions proved by
the authors.
