On Euler's formulae for double zeta values

2016/04/15 by Ryotaro Harada, Harada, Ryotaro
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1604.04463

openalex publication_date 2016/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• A signed analog of Euler's reduction formula for the double zeta function
• Weighted and Restricted Sum Formulas of Euler Sums
• On certain mean values of the double zeta-function
• Alternating Double Euler Sums, Hypergeometric Identities and a Theorem of Zagier
• On the recurrence formula of the Euler zeta functions