Venue: Microsoft Teams (online)

Multiple zeta values are the real numbers \begin{equation} \zeta({\bf a})= \sum_{n_1>\cdots>n_r>0}n_1^{-a_1}\cdots n_r^{-a_r}, \end{equation} where ${\bf a}=(a_1, \ldots ,a_r)$ is an admissible composition, i.e. a finite sequence of positive integers, with $a_1 \geqslant 2$ when $r\neq 0$.

The multiple Apéry-like sums defined by \begin{equation} \sigma({\bf a})=\sum_{n_1>\cdots>n_r>0}\left({2 n_1 \atop n_1}\right)^{-1}n_1^{-a_1}\cdots n_r^{-a_r} \end{equation} when ${\bf a}\neq\varnothing$ and by $\sigma(\varnothing)=1$. We show that for any admissible composition ${\bf a}$, there exists a finite formal $\bf Z$-linear combination $\sum \lambda_{\bf b} {\bf b}$ of admissible compositions such that \begin{equation} \zeta({\bf a})=\sum \lambda_{\bf b}\, \sigma({\bf b}). \end{equation} The simplest instance of this fact is the identity \begin{equation} \sum_{n=1}^{\infty}\frac{1}{n^2}=3\sum_{n=1}^{\infty}\frac{1}{\left({2n \atop n}\right)n^2} \end{equation} discovered by Euler, which expresses that $\zeta(2)=3\,\sigma(2)$. Note that multiple Apéry-like sums have the advantage on multiple zeta values to be exponentially quickly convergent.

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 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 $\zeta(2,\ldots,2,3,2,\ldots,2)$ (D. Zagier, Evaluation of the multiple zeta values $\zeta(2,\ldots,2,3,2,\ldots,2)$, Annals of Math. 175 (2012), 977–1000) as well as of similar ones in the context of Apéry-like sums.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 16 Jan 2022