On Euler's formulae for double zeta values

TL;DR

This paper rigorously reformulates Euler's classical formulae for double zeta values, providing proofs and connecting them to modern extended double shuffle relations, thus clarifying their mathematical foundations.

Contribution

It offers a precise reformulation and rigorous proof of Euler's double zeta formulae, linking them to extended double shuffle relations for the first time.

Findings

Euler's formulae are now rigorously proven.

The formulae can be derived from extended double shuffle relations.

Clarification of the mathematical completeness of Euler's methods.

Abstract

In 1776, L. Euler proposed three methods, called prima methodus, secunda methodus and tertia methodus, to calculate formulae for double zeta values. However strictly speaking, his last two methods are mathematically incomplete and require more precise reformulation and more sophisticated arguments for their justification. In this paper, we reformulate his formulae, give their rigorous proofs and also clarify that the formulae can be derived from the extended double shuffle relations.