Another look at Zagier's formula for multiple zeta values involving 2's and 3's.

In this talk, we shall discuss about Zagier's formula for the multiple zeta values, $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ and its connections to Brown's proofs of the conjecture on the Hoffman basis and the zig-zag conjecture of Broadhurst in quantum field theory. Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd. 

By using the Taylor series of integer powers of arcsin function and a related result about expressing rational zeta series involving $\zeta(2n)$ as a finite sum of $\mathbb{Q}$-linear combinations of odd zeta values and powers of $\pi$, we derive a new and direct proof of Zagier's formula in the special case $\zeta(2, 2, \ldots, 2, 3)$.