Crm Talk and Handout

This talk and papers concern sums of the type (a; b; c) := 1 X x=1 x 1 X y=1 y 1 X z=1 1 xaybzc ; and their n-variable analogues and polylogarithmic extensions. For which integers a; b; c are these sums reducible: expressible in terms of the simpler series (a; b) := 1 X x=1 x 1 X y=1 1 xayb and (a) := 1 X x=1 1 xa ? Sums of this type are (triple, double or single) Euler sums or Multi-dimensional zeta function values (MZVs), as Euler was the rst to study them and, of course, the single Euler sums are integral values of the Riemann zeta function.