On the Laurent series for the Epstein zeta function

The Epstein zeta function zeta(d)[M; s] equivalent to Sigma(n is an element of Zd)'(nM n(T))(-s/2), where M is a real symmetric and invertible d x d matrix and n is a d-dimensional row vector (n(1), n(2),..., n(d)) with integer coordinates n(i), is considered. (The prime on the sum indicates that the term n = 0 should be excluded.) It is known that zeta(d) [M; s] has a Laurent series expansion about the singular point s = d which can be written in the form zeta(d)[M; s] = Sigma(infinity)(nu=-1) A(nu) [M; d] (s - d)(nu). In this paper we shall show that the coefficient A(nu) [M; d] can be accurately calculated using rapidly convergent series which involve the Meijer G-function. Exact formulae are also derived for A(nu)[M; 2] when M = U-N equivalent to (1 0 0 N), with N = 1, 2, .... The results for A(0)[UN; 2] are then used to establish several mathematical identities involving summations of generalized Stieltjes constants. Next the Laurent series for zeta(d)[I-d; s], where I-d is the d x d unit matrix, is briefly discussed for the cases d = 3, 4, 6 and 8. Finally, a new application of the results in lattice statistics is described.