Weyl asymptotics of Bisingular Operators and Dirichlet Divisor Problem

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.