Introduction of an Elementary Method to Express $\zeta(2n+1)$ in Terms of $\zeta(2k)$ with $k\geq 1$

TL;DR

This paper presents a simple, elementary method based on Euler's work to express odd zeta values in terms of even zeta values, making the concept more accessible to learners.

Contribution

It introduces the most elementary known approach to relate b6(2n+1) to b6(2k) for k  1, using only basic Eulerian techniques.

Findings

Provides a straightforward method for expressing b6(2n+1) in terms of b6(2k)

Uses only elementary mathematical tools, making it accessible to students

Enhances understanding of the relationship between odd and even zeta values

Abstract

In this note we give the most elementary method (as far as we know) to express $\zeta(2n+1)$ in terms of $\{\zeta(2k)|k\geq 1\}$. The method is based on only some elementary works by Leonhard Euler, so it is very instructive to non-experts or students.