  We study the general theory of weighted Dirichlet series and associated
summatory functions of their coefficients. We show that any non-real pole leads
to oscillatory error terms. This applies even if there are infinitely many
non-real poles with the same real part. Further, we consider the case when the
non-real poles lie near, but not on, a line. The method of proof is a
generalization of classical ideas applied to study the oscillatory behavior of
the error term in the prime number theorem.
