An elementary and real approach to values of the Riemann zeta function

TL;DR

This paper introduces an elementary method to compute the Riemann zeta function at negative integers without relying on complex analysis, using a novel ordering of integers and summation of divergent series.

Contribution

It presents a new elementary approach to evaluate the Riemann zeta function at negative integers, bypassing traditional complex analysis techniques.

Findings

Values at negative integers computed without analytic continuation

New integer ordering method for summation

Alternative summation of divergent series

Abstract

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that the values of the Riemann zeta function can be computed, without using the theory of analytic continuation and functions of complex variable.