Epstein zeta-function

Epstein -function

A function belonging to a class of Dirichlet series generalizing the Riemann zeta-function (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 -matrix and , the Epstein zeta-function is defined by

where stands for the transpose of . The series converges absolutely for . If and , it equals .

The Epstein zeta-function shares many properties with the Riemann zeta-function (cf. [a5], V.Sect. 5, [a8], 1.4, [a9]):

possesses a meromorphic continuation to the whole -plane (cf. also Analytic continuation) with two simple poles, at and , and satisfies the functional equation

Thus, is holomorphic in except for a simple pole at with residue

Moreover, one has

It should be noted that the behaviour may be totally different from the Riemann zeta-function. For instance, for there exist matrices such that has infinitely many zeros in the half-plane of absolute convergence (cf. [a1]), respectively a zero in any point of the real interval (cf. [a8], 4.4).

The Epstein zeta-function is an automorphic form for the unimodular group (cf. [a8], 4.5), i.e.

It has a Fourier expansion in the partial Iwasawa coordinates of involving Bessel functions (cf. [a8], 4.5). For 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 in an -dimensional Euclidean vector space . One has

where is the Gram matrix of the basis .

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 attached 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)