Multidimensional extension of the generalized Chowla-Selberg formula

After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an expo nentially convergent expression for the analytic continuation of a multidim ensional inhomogeneous Epstein-type zeta function of the general form zeta(A,(b) over right arrow,q)(s) = Sigma((n) over bar is an element of Zp) ((n) over right arrow(T) A (n) over bar + (b) over right arrow(T) (n) over right arrow + q)(-s), with A the p x p matrix of a quadratic form, (b) over right arrow a p vecto r and q a constant, is obtained. It is valid on the whole complex s-plane, is exponentially convergent and provides the residua at the poles explicitl y. it reduces to the famous formula of Chowla and Selberg in the particular case p = 2, (b) over right arrow = (0) over right arrow, q = 0, Some varia tions of the formula and physical applications are considered.