Summation and the Poisson formula

TL;DR

This paper introduces an algebraic definition of series summation for functions like the Riemann zeta and Epstein zeta functions, demonstrating their intrinsic meaning as complex numbers independent of analytic continuation, using the Poisson formula.

Contribution

It provides an algebraic approach to summing series related to important functions, clarifying their meaning without relying on analytic continuation.

Findings

Series sums have an intrinsic algebraic meaning as complex numbers.

The approach applies to the Riemann zeta and Epstein zeta functions.

The Poisson formula links algebraic sums to analytical significance.

Abstract

By giving the definition of the sum of a series indexed by a set on which a group acts, 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 $\Z^k$ 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.