Resolution of some multiple zeta conjectures
by
Douglas Bowman
University of Illinois
Coauthors: David Bradley (University of Maine)

We resolve three conjectured arbitrary depth multiple zeta evaluations. These identities have their origin in a 1994 paper of Don Zagier in which he presented a conjectured evaluation of an n-dimensional zeta sum. This problem remained open until recently when it was resolved by the physicist David Broadhurst with the assistance of Jon Borwein and David Bradley (Trans. AMS, to appear). Zagier later simplified the proof, but the final Broadhurst-Zagier proof relied in one key step on computer algebra. In the paper settling the Zagier conjecture as well as in another paper, Borwein, Bradley and Broadhurst presented a number of new conjectures which their methods were unable to handle. These conjectures were discovered through extensive computer searches using lattice basis reduction.

Using a new technique, we settle three of these conjectures and give several new results which were missed in the computer searches. Our method removes the need for computer algebra in the Broadhurst-Zagier proof and replaces it with a proposition about differential fields. Our solution also makes use of asymptotic estimates at singular points of hypergeometric functions.

Date received: March 23, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-53.