The ACO Seminar (2017–2018)

The multiple zeta values (Euler–Zagier sums) were introduced independently by Hoffman and Zagier in 1992 and they play a crucial role at the interface between analysis, number theory, combinatorics, algebra and physics.

The central part of the talk is given by Zagier's formula for the multiple zeta values, ζ(2, 2,..., 2, 3, 2, 2,..., 2). 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 ζ(2, 2,..., 2, 3, 2, 2,..., 2) as rational linear combinations of products ζ(m)π²ⁿ with m odd. 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.

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

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.