# Algebra/Number Theory Seminar

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

Cezar Lupu, Beijing Institute (IMSA),Yau Center,Tsing-Hua U

Thursday, February 24th, 2022
11:00 AM - 12:00 PM
Zoom id 566 385 6457 (no password)

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.

Event contact: dinesh dot thakur at rochester dot edu