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 $P \in \R [X_1,..., X_n]$ has degree $d$ 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.
Categories: math.NT math.AG