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Multiple zeta values and multiple Apéry-like sums. (English) Zbl 1479.11147

Summary: In this paper, we formally introduce the notion of Apéry-like sums and we show that every multiple zeta values can be expressed as a \(\mathbb{Z}\)-linear combination of them. We even describe a natural way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier’s formulas for \(\zeta(2,\dots,2,3,2,\dots,2)\) as well as of similar ones in the context of Apéry-like sums. The relations between Apéry-like sums themselves still remain rather mysterious, but we get significant results and state some conjectures about their pattern.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
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