Mixed zeta functions and application to some lattice points problems

Details Mixed zeta functions and application to some lattice points problems Mixed zeta functions and application to some lattice points problems

2005-05-26

arxiv.org/abs/math/0505558v2

Mathematics

Number Theory

26 pages

Scientific paper

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.

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