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 exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form ζA,~b,q(s) = ∑ ~n∈Z (~nA~n+~b~n+ q)−s, with A the p× p matrix of a quadratic form, ~b a p vector 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 explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case p = 2, ~b = ~0, q = 0. Some variations of the formula and physical applications are considered. PACS: 11.10.Gh, 02.30.Tb, 02.30.Dk, 02.30.Mv E-mail address: [email protected], [email protected] 1