In this talk, we explore Zagier's famous formula for multiple zeta values involving 2's and 3's.

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.

In Zagier's paper (http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p11-p...), the formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality.

Last but not least, we give an insight how this formula served as a key ingredient in solving the famous zig-zag conjecture of Broadhurst and Kreimer in quantum field theory.

Thackeray 704