Explicit formulas are given for the analytic continuation of multiple-sum zeta functions of the general ‘diagonal’ Epstein type plus mass term: \[ \sum_{l_ 1, \dots, l_ N= -\infty}^ \infty \bigl[ a_ 1 (l_ 1- c_ 1)^ 2+ \cdots+ a_ N (l_ N- c_ N)^ 2+ M^ 2 \bigr]^{- s}, \] with \(a_ i>0\), \(i=1,2,\dots, N\), \(M^ 2\geq 0\), in terms of series of Riemann and Hurwitz zeta functions, and of Kelvin (modified Bessel) functions. Using different analytic continuation methods several formulas are obtained which are specifically useful for different ranges of values of the involved parameters.
As noted by the author himself, using the definition of functional determinants through the zeta function of the corresponding differential operator, these formulas are very useful for explicit calculations in several domains of quantum physics, such as in the study of dimensional reduction and spontaneous compactification.
What renders the article interesting is the fact that such kind of general expressions are difficult to find in the literature. In one of the versions given in the paper, the final formulas involve multiple derivatives of a basic expression in which the \(c_ i\)’s are absent. In another version, the formulas coincide with the ones derived in the reviewer’s paper [J. Math. Phys. 31, 170-174 (1990; Zbl 0753.11031)].
For a different, more general treatment involving arbitrary exponents see the reviewer [J. Phys. A 22, 931-942 (1989; Zbl 0699.10061)], for the generalization to non-diagonal, inhomogeneous Epstein zeta functions see the reviewer [J. Phys. A 27, 3775-3785 (1994) (an explicit generalization of the Chowla-Selberg formula is given)], and for a careful numerical treatment of some of the equations in the paper see the reviewer [J. Math. Phys., Sept. 94 (to appear)].