BIMSA >
BIMSA-YMSC Tsinghua Number Theory Seminar
Elementary proofs of Zagier's formula for multiple zeta values and its odd variant
Elementary proofs of Zagier's formula for multiple zeta values and its odd variant
Organizers
Hansheng Diao
, Yueke Hu
, Emmanuel Lecouturier
,
Cezar Lupu
Speaker
Li Lai
Time
Tuesday, July 12, 2022 4:00 PM - 5:00 PM
Venue
1110
Online
Zoom 361 038 6975
(BIMSA)
Abstract
In 2012, Zagier proved a formula which expresses the multiple zeta values
\[ H(a, b)=\zeta(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}) \]
as explicit $\mathbb{Q}$-linear combinations of products $\pi^{2m}\zeta(2n+1)$ with $2m+2n+1=2a+2b+3$. Recently, Murakami proved an odd variant of Zagier's formula for the multiple $t$-values
\[ T(a, b)=t(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}). \]
In this talk, we will give new and parallel proofs of these two formulas. Our proofs are elementary in the sense that they only involve the Taylor series of powers of arcsine function and certain trigonometric integrals. Thus, these formulas become more transparent from the view of analysis. This is a joint work with Cezar Lupu and Derek Orr.