Series representation of arborified zeta values

Abstract

We show that any convergent (shuffle) arborified zeta values admits a series representation. As a consequence we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formula. The series representation of shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multizeta values with rational coefficients.

Pole structure of Shintani zeta functions and Newton polytopes

“…Riemann proved in [R] that ζ admits a meromorphic continuation to the whole complex plane with a simple pole at s = 1. Several multivariable generalisations of the Riemann zeta functions have been made, for instance the multiple zeta functions (or Euler-Riemann-Zagier zeta functions) [Za], [Z] also called poly zeta functions [C], Shintani zeta functions [GPZ1], [M1], conical zeta functions [GPZ1], [GPZ2], [CGPZ1], Mordell-Tornheim zeta functions [M1], [M2], branched or arborified zeta functions [CGPZ2,Cl1,Cl2], etc. Meromorphic continuations of such generalisations have called the attention of numerous mathematicians [AET], [M1], [M2], [Z].…”