Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Volumes, Traces and Zeta Functions   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.

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The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th