Epstein zeta-function

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 [a4] in 1903 after special cases had been dealt with by L. Kronecker [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. [a5], V.Sect. 5, [a8], 1.4, [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. [a1]), respectively a zero in any point of the real interval $(0,n/2)$ (cf. [a8], 4.4).

The Epstein zeta-function is an automorphic form for the unimodular group $\mathrm{GL}_n(\mathbf{Z})$ (cf. [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. [a8], 4.5). For $n=2$ it coincides with the real-analytic Eisenstein series on the upper half-plane (cf. Modular form; [a5], V.Sect. 5, [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. [a7]). In the case of an arbitrary algebraic number field it gives an integral representation of the associated Dedekind zeta-function (cf. [a8], 1.4).

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

References

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