On Euler's Formulae for Double Zeta Values

Abstract

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, provide their rigorous proofs and also clarify that the formulae can be derived from the extended double shuffle relations. Euler's methodsIn 1776, L. Euler published a celebrated paper Meditations circa singulare serierum genus † in Latin, which translates to 'Meditations about a singular type of series' in English. It is said to be the first publication in history where multiple zeta values (actually only double zeta values) were introduced. In the paper he proposed three methods to calculate certain relations among double zeta values, which he called prima methodus, secunda methodus and tertia methodus. Here we explain the method behind his ideas and point out the steps that would be considered insufficient. Prima methodusIn his paper, he studied the serieswhich he denoted by the unconventional notation (1/z m )(1/y n ) and also the series 1 + 1/2 m + 1/3 m + 1/4 m + · · · which he again denoted by the notation 1/z m . In modern language, they are nothing but the double zeta star valuefor n ∈ Z >0 and m ∈ Z >1 , and the Riemann zeta value ζ (m) := ∞ k=1 1/k m for m ∈ Z >1 , respectively. Here we recall the double zeta value ζ (n, m) :=

Euler–zagier Sums via Trigonometric Series

“…In other words, if the conjecture on Euler sums of even weight that we mentioned before is true for some H(p, q) with an even weight w = p + q ≥ 8, then there is another pair (p 1 , q 1 ) where p 1 + q 1 ≤ w, p 1 / ∈ {p, q} and the conjecture also holds for H(p 1 , q 1 ). The following result may also follow from what Euler did, and for rigorous proofs for Euler's results, we direct the reader to [5].…”

“…Now we proceed by induction on weight w = 2r + 4. If we take r = 2 in Equation (7.5), one sees that −48H(2, 6) = ZV 7 + 12H(2, 6) + 24H (3,5). This yields that H(2, 6) ∈ Ω if and only if H(3, 5) ∈ Ω.…”

On Lara Rodríguez’ full conjecture for double zeta values in function fields

“…Allegedly, in 1776, the double zeta values (multiple zeta values with depth 2) were firstly introduced by L. Euler in [4] where he also described three types of relations for double zeta values with non-mathematical proofs and unconventional notations (they were reformulated with mathematical proofs and conventional modern notations in [5]). It is said that the multiple zeta values were rediscovered after the silence of more than two centuries.…”