Combinatorial Aspects of Multiple Zeta Values

Abstract

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

Massive 3-loop Feynman diagrams reducible to SC $^*$ primitives of algebras of the sixth root of unity

“…in terms of Saalschützian hypergeometric series of the form [1] W (µ, ν; α, β) := 3 F 2 ( 1 2 − µε, 1 2 − νε, 1; 3 2 + αε, 3 2 + βε; 1) ( 1 2 + αε)( 1 2 + βε) (13) constrained by wreath-product symmetry. They generate polylogarithmic ladders [6] that enable us to compute the ten millionth hexadecimal digits of the QCD constants ζ(3) and ζ (5).…”

“…one may transform the integrand to products of log(x), Li 0 (x) := x/(1 − x), Li 1 (x) := − log(1 − x) and Li 2 (x) := x 0 (dy/y) Li 1 (y). Then the expansion Li k (x) = n>0 x n /n k makes the integration trivial and produces non-alternating Euler sums of weight 4, all of which are proven [13] to evaluate to rational multiples of π 4 , with 1 90 π 4 = ζ(4) = ζ(2, 1, 1) = 4ζ(3, 1) = 4 3 ζ(2, 2)…”

“…Thus we had a very strong expectation that the same simplification would occur in the present case. Indeed it does; using the methods of [7,13] we evaluated (50) as…”

Weight Systems from Feynman Diagrams

“…In the limit of ρ → 0 we have < I 1,1 + I 1,3 + I 10,1 + I 10,3 > = 0 (Ia), < I 5,1 + I 5,3 + I 6,1 + I 6,3 > = 0 (Ib), < I 2,1 + I 2,3 + I 9,1 + I 9,3 > = 0 (IIa), 12 A different proof may be obtained by isolating the singularity at u = 0 in appropriate functions f 2 or f 4 . At these points, the paired integrands match and cancel out.…”

A quantum field theoretical representation of Euler‐Zagier sums

“…However, we have not been able to locate in the literature an exact equivalent of our length m identities (4.25),(4.28) 3 . The only identities available for arbitrary lengths and levels are those based on the 'shuffle algebra' [6,26,10,7,8] and its generalizations [23]. Of those the 'depth-length' shuffle identities (which are also called 'stuffle identities' or ' * products' [22]) are obviously related, and in fact equivalent to our 'permutation' identities, as we have convinced ourselves.…”