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

Weyl asymptotics of bisingular operators and Dirichlet divisor problem