Volumes, Traces and Zeta Functions

Abstract

Let $Q(x)$ be a quadratic form over $\mathbb{R}^n$. The Epstein zeta function associated to $Q(x)$ is a well known function in number theory. We generalize the construction of the Epstein zeta function to a class of function $\phi(x)$ defined in $\mathbb{R}^n$ that we call $A-$homogeneous, where $A$ is a real aquare matrix of order $n$ having each eigenvalue in the left hal space $\Re\lambda>0$. Such a class includes all the homogeneous polynomials (positive outside the origin) and all the norms on $\mathbb{R}^n$ which are smooth outside the origin. As in the classical (i.e. quadratic) case we prove that such zeta functions are obtained from the Mellin transforms of theta function of Jacobi type associated to the $A-$homogeneous function $\phi(x)$. We prove that the zeta function associated to a $A-$homogeneous function $\phi(x)$ which is positive and smooth outside the origin is an entire meromorphic function having a unique simple pole at $s=\alpha$ the trace of the matrix $A$ with residue given by the product of the trace $\alpha$ and the Lebesgue volume of the unit ball associated to $\phi(x)$, that is the volume of the set $x\in\R^n$ satisfying $\phi(x)<1$. We also prove that the theta funtion associated to $\phi(x)$ has an asymptotic expansion near the origin. We find that the coefficients of such expansion depend on the values that the zeta function associated to $\phi(x)$ assumes at the negative integers.