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