Zagier’s formula for multiple zeta values and its odd variant revisited

Zoom Meeting ID: 973 0230 7263

Thursday, April 21, 2022 - 11:00
Speaker Information
Cezar Lupu
Beijing Institute of Mathematical Sciences and Applications (BIMSA) & Yau Mathematical Sciences Center (YMSC), Tsinghua University

Abstract or Additional Information

In this talk, we revisit the famous Zagier formula for multiple zeta values (MZV's) and its odd variant for multiple $t$-values which is due to Murakami. Zagier's formula involves a specific family of MZV's which we call nowadays the Hoffman family, $$\displaystyle H(a, b)=\zeta(\underbrace{2, 2, \ldots, 2}_{\text{$a$}}, 3, \underbrace{2, 2, \ldots, 2}_{\text{$b$}}),$$
which can be expressed as a $\mathbb{Q}$-linear combination of products $\pi^{2m}\zeta(2n+1)$ with $m+n=a+b+1$. This formula for $H(a, b)$ played a crucial role in the proof of Hoffman's conjecture by F. Brown, and it asserts that all multiple zeta values of a given weight are $\mathbb{Q}$-linear combinations of MZV's of the same weight involving $2$'s and $3$'s. 
 Similarly, in the case of multiple $t$-values (the odd variant of multiple zeta values), very recently, Murakami proved a version of Brown's theorem (Hoffman's conjecture) which states that every multiple zeta value is a $\mathbb{Q}$-linear combination of elements  $\{t(k_{1}, \ldots, k_{r}): k_{1}, \ldots, k_{r}\in \{2, 3\}\}$. Again, the proof relies on a Zagier-type evaluation for the Hoffman's family of multiple $t$-values, $$\displaystyle T(a, b)=t(\underbrace{2, 2, \ldots, 2}_{\text{$a$}}, 3, \underbrace{2, 2, \ldots, 2}_{\text{$b$}}).$$
We show the parallel of the two formulas for $H(a, b)$ and $T(a, b)$ and derive elementary proofs by relating both of them to a surprising cotangent integral. Also, if time allows, we give a brief account on how these integrals can provide us with some arithmetic information about $\frac{\zeta(2k+1)}{\pi^{2k+1}}$. This is a joint work with Li Lai and Derek Orr.

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