Mixed zeta functions and application to some lattice points problems

We consider zeta functions: $Z(f ;P ;s)=\sum_{\m \in \N^{n}} f(m_1,…, m_n) P(m_1,…, m_n)^{-s/d}$where$P \in \R [X_1,…, X_n]$hasdegree$d$and$f$ is a function arithmetic in origin, e.g. a multiplicative function. In this paper, I study the meromorphic continuation of such series beyond an a priori domain of absolute convergence when $f$ and $P$ satisfy properties one typically meets in applications. As a result, I prove an explicit asymptotic for a general class of lattice point problems subject to arithmetic constraints.