Summation and the Poisson formula

Abstract

By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by $${\mathbb {Z}}^k$$ has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.