  We consider a class of pseudodifferential operators defined on the product of
two closed manifolds, with crossed vector valued symbols. We study the
asymptotic expansion of Weyl counting function of positive selfadjoint
operators in this class. Exploiting a general Theorem of J. Aramaki, we
determine the first term of the asymptotic expansion of Weyl counting function
and, in a special case, we find the second term. We finish with some examples,
emphasizing connections with problems of analytic number theory, in particular
with Dirichlet Divisor Problem.
