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''Epstein $\zeta$-function''

A function belonging to a class of [[Dirichlet series]] generalizing the [[Riemann zeta-function]] $\zeta(s)$ (cf. also [[Zeta-function]]). It was introduced by P. Epstein [[#References|[a4]]] in 1903 after special cases had been dealt with by L. Kronecker [[#References|[a6]]], IV, 495. Given a real positive-definite $n\times n)$-matrix $T$ and $s \in \mathbf{C}$, the Epstein zeta-function is defined by
$$
\zeta(T;s) = \sum_{\mathbf{0} \ne g \in \mathbf{Z}^n} (g^\top T g)^{-s}
$$
where $g^\top$ stands for the transpose of $g$. The series converges absolutely for $\mathrm{re} s > n/2$. If $n=1$ and $T=(1)$, it equals $2\zeta(2s)$.

The Epstein zeta-function shares many properties with the Riemann zeta-function (cf. [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 1.4, [[#References|[a9]]]):
$$
\xi(T;s) = \pi^{-s} \Gamma(s) \zeta(T;s)
$$
possesses a meromorphic continuation to the whole $s$-plane (cf. also [[Analytic continuation]]) with two simple poles, at $s = n/2$ and $s=0$, and satisfies the functional equation
$$
\xi(T;s) = (\det T)^{-1/2} \xi\left({ T^{-1};\frac{n}{2}-s }\right) \ .
$$

Thus, $\zeta(T;s)$ is holomorphic in $s \in \mathbf{C}$ except for a simple pole at $s=n/2$ with residue
$$
\frac{\pi^{n/2}}{ \Gamma(n/2)\sqrt{\det T} } \ .
$$

Moreover, one has
$$
\zeta(T;0) = -1
$$
$$
\zeta(T;-m) = 0\ \ \text{for}\ \ m=1,2,\ldots \ .
$$

It should be noted that the behaviour may be totally different from the Riemann zeta-function. For instance, for $n>1$ there exist matrices $T$ such that $\zeta(T;s)$ has infinitely many zeros in the half-plane of absolute convergence (cf. [[#References|[a1]]]), respectively a zero in any point of the real interval $(0,n/2)$ (cf. [[#References|[a8]]], 4.4).

The Epstein zeta-function is an [[automorphic form]] for the [[unimodular group]] $\mathrm{GL}_n(\mathbf{Z})$ (cf. [[#References|[a8]]], 4.5), i.e.
$$
\zeta(U^\top T u;s) = \zeta(T;s) \ \ text{for}\ \ U \in \mathrm{GL}_n(\mathbf{Z}) \ .
$$

It has a Fourier expansion in the partial Iwasawa coordinates of $T$ involving [[Bessel functions]] (cf. [[#References|[a8]]], 4.5). For $n=2$ it coincides with the real-analytic [[Eisenstein series]] on the upper half-plane (cf. [[Modular form]]; [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 3.5).

The Epstein zeta-function can also be described in terms of a lattice $\Lambda = \mathbf{Z}\lambda_1 + \cdots + \mathbf{Z}\lambda_n$ in an $n$-dimensional Euclidean vector space $(V,\sigma)$. One has
$$
\zeta(T;s) = \sum_{0 /ne \lambda \in \Lambda} \sigma(\lambda,\lambda)^{-s} \ ,
$$
where $T = (\sigma(\lambda_i,\lambda_j))$ is the [[Gram matrix]] of the basis $\lambda_1,\ldots,\lambda_n$.

Moreover, the Epstein zeta-function is related with number-theoretical problems. It is involved in the investigation of the  "class number one problem"  for imaginary quadratic number fields (cf. [[#References|[a7]]]). In the case of an arbitrary algebraic [[number field]] it gives an integral representation of the associated [[Dedekind zeta-function]] (cf. [[#References|[a8]]], 1.4).

The Epstein zeta-function plays an important role in crystallography, e.g. in the determination of the Madelung constant (cf. [[#References|[a8]]], 1.4). Moreover, there are several applications in mathematical physics, e.g. [[quantum field theory]] and the Wheeler–DeWitt equation (cf. [[#References|[a2]]], [[#References|[a3]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Davenport,   H. Heilbronn,   "On the zeros of certain Dirichlet series I, II"  ''J. London Math. Soc.'' , '''11'''  (1936)  pp. 181–185; 307–312</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Elizalde,   "Ten physical applications of spectral zeta functions" , ''Lecture Notes Physics'' , Springer  (1995)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Elizalde,   "Multidimensional extension of the generalized Chowla–Selberg formula"  ''Comm. Math. Phys.'' , '''198'''  (1998)  pp. 83–95</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Epstein,   "Zur Theorie allgemeiner Zetafunktionen I, II"  ''Math. Ann.'' , '''56/63'''  (1903/7)  pp. 615–644; 205–216</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Koecher,   A. Krieg,   "Elliptische Funktionen und Modulformen" , Springer  (1998)</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Kronecker,   "Werke I—V" , Chelsea  (1968)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Selberg,   Chowla, S.,   "On Epstein's Zeta-function"  ''J. Reine Angew. Math.'' , '''227'''  (1967)  pp. 86–110</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Terras,   "Harmonic analysis on symmetric spaces and applications" , '''I, II''' , Springer  (1985/8)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  E.C. Titchmarsh,   D.R. Heath–Brown,   "The theory of the Riemann zeta-function" , Clarendon Press  (1986)</TD></TR>
</table>

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