# Witten multiple zeta values attached to

###### Key words and phrases:

Witten multiple zeta functions, multiple zeta values.###### 1991 Mathematics Subject Classification:

Primary: 11M41; Secondary: 40B05Jianqiang Zhao

Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, USA

Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Xia Zhou

Department of Mathematics, Zhejiang University, Hangzhou, P.R. China, 310027

Abstract. In this paper we shall prove that every Witten multiple zeta value of weight attached to at nonnegative integer arguments is a finite -linear combination of MZVs of weight and depth three or less, except for the nine irregular cases where the Riemann zeta value and the double zeta values of weight and depth are also needed.

## 1. Introduction

It is well-known that suitably defined zeta and -functions and their special values often play significant roles in many areas of mathematics. In [21] Witten studied one variable zeta functions attached to various Lie algebras and related their special values to the volumes of certain moduli spaces of vector bundles of curves. Zagier [22] (and independently Garoufalidis) gave direct proofs that such functions at positive even integers are rational multiples of powers of . More recently Matsumoto and his collaborators [12, 13, 14, 15, 17] defined multiple variable versions of these functions and began to investigate their analytical and arithmetical properties.

Let be the set of positive integers and . For any
we let be a poset with the usual
increasing order. By
we mean a nonempty subset of
as a poset. We say the length of is
and the weight of is . We define
the *generalized multiple zeta function* of depth as

(1) |

For example, the Euler-Zagier multiple zeta function [1, 22, 23]

(2) |

corresponds to the special case that unless for . The Mordell-Tornheim multiple zeta function defined by (10) (see [18, 20]) is the case where unless or . If we set unless is a consecutive string of positive integers then we get exactly Witten multiple zeta function associated to the special linear Lie algebra (see [17]):

(3) |

The generalized multiple zeta-functions defined by (1)
are special cases of the functions studied by Essouabri [6], de Crisenoy [5],
and Matsumoto [16]. In particular we know that
has meromorphic continuation to the whole complex space .
However, in the form (1) we may have better
control of its arithmetical properties, namely, we may be able
to compute explicitly their special values at nonnegative integers.
As usual we say a value of the
Euler-Zagier multiple zeta function at positive integers
a *multiple zeta value* (MZV for short) if it is finite. Our
major interest is to solve the following problem.

Main Problem. Suppose (resp. ) and converges. Is always a -linear combination of MZVs of the same weight (resp. same or lower weights) and of depth or less?

When the function in (1) becomes Mordell-Tornheim double zeta function. By the main result of [24] (see Prop. 2.4) we know the above Main Problem has an affirmative answer for all Mordell-Tornheim multiple zeta functions. In this paper we will consider which is essentially the case when .

Throughout the rest of the paper whenever we say some special value with positive (resp. nonnegative) integer arguments is expressible by MZVs we mean that the value can be expressed as a finite -linear combination of MZVs of the same (resp. same or lower) weights and the same or lower depths. Our main result is the following theorem which provides an affirmative answer to the above Main Problem for the case .

###### Theorem 1.1.

Suppose (resp. ). If converges then it is expressible by MZVs. Moreover, very Witten multiple zeta value of weight attached to at nonnegative integers is a -linear combination of MZVs of weight and depth three or less, except for the nine irregular cases defined by (13) to (17) where the Riemann zeta value and the double zeta values of weight and depth are also needed.

The first author wishes to thank Max-Planck-Institut für Mathematik for providing financial support during his sabbatical leave when this work was done. The second author is supported by the National Natural Science Foundation of China, Project 10871169.

## 2. Some preliminary results

In this section we collect some useful facts which will be convenient for us to present our main result in later sections.

### 2.1. Convergence domain

We assume that all components of are integers and derive the necessary and sufficient conditions for (1) to converge although the conditions are still sufficient if we allow complex variables and take the corresponding real parts of the variables in the conditions.

Recall that the MZV in (2) converges if and only if

(4) |

for all . It is straight-forward to see the same holds for the “star” version of the multiple zeta function:

(5) |

Special values of (5) were studied in [9] and [19]. We can extend this convergence criterion easily to our generalized multiple zeta functions.

###### Proposition 2.1.

The generalized MZV

(6) |

converges if and only if for all and all

(7) |

###### Proof.

The idea of the proof is similar to that of [24, Thm. 4]. First we observe that for any subset we have

Hence

(8) |

where LHS means the quantity at the extreme left of the above inequalities. Observe that for each fixed the power of in LHS is where runs through all subsets of whose first component is . Hence the criterion (4) implies that LHS and (LHS) of (8) converges if and only if for each fixed

for all . Let (). Then in the above sum runs through all subset of containing at least some (). This is exactly (7), as desired. ∎

###### Remark 2.2.

It is easy to see that similar result holds if we replace the factors in (6) by linear forms of with nonnegative integer coefficients.

### 2.2. MZVs with arbitrary integer arguments

In this paper we will mostly consider multiple zeta values with nonnegative integer arguments as long as they converge. However, we will prove a more general result as follows since the inductive proof forces us to do so.

###### Proposition 2.3.

Suppose . If converges then it can be expressed as a -linear combination of MZVs (at positive integer arguments) of the same or lower weights and the same or lower depths.

###### Proof.

We prove the proposition by induction on the depth. When we have nothing to prove. Suppose the proposition holds for all MZVs of depth . Suppose further

(9) |

so that converges. Assume . Then by definition

Now by the well-known formula (see, for e.g., [11, p. 230, Thm. 1])

where is the Bernoulli polynomial we immediately see that is a -linear combination of MZVs of the forms and where . All of these MZVs are easily to be shown as convergent values by (9) and therefore the proposition follows from induction assumption. ∎

### 2.3. Mordell-Tornheim zeta functions

###### Proposition 2.4.

*([24, Thm. 5])*
Let and be nonnegative integers. If at most
one of them is equal to then the Mordell-Tornheim zeta value
can be expressed as a -linear
combination of MZVs of the same weight and depth.

In this paper we will only need this proposition when the depth is three.

### 2.4. A combinatorial lemma

The next lemma will be used heavily throughout the paper.

###### Lemma 2.5.

*([24, Lemma 1])*
Let and be positive integers, and let
be non-zero real numbers such that .
Then

where the multi-nomial coefficient

The notation means the multiple sum

## 3. Proof of Theorem 1.1

We now use a series of reductions to prove the theorem. All of the steps will be explicitly given so that one may carry out the computation of (11) by following them.

Step (i). Since we see clearly that we can assume either or or . In fact in Lemma 2.5 taking , , , , , and we get:

We recommend the interested reader to check the convergence of the above values by (12). The rule of thumb is that if we apply Lemma 2.5 with each a positive combination of indices then the convergence is automatically guaranteed. In each of the following steps we often omit this convergence checking since it is straight-forward in most cases. The only exception is (26) which in fact poses the most difficulty.

By symmetry we see that without loss of generality we only need to show that is expressible by MZVs. This is nothing but the Matsumoto’s version of Witten multiple zeta function of depth 3 associated to the special linear Lie algebra (see (3)):

Before going on we need to define the so called *regular* and *irregular*
special values of .
Clearly the following special values are expressible by MZVs of
mixed weights: ( means to repeat times)

(13) | |||

(14) | |||

(15) |

By Step (ii) we will see that special values

(16) |

are also expressible by MZVs of mixed weights. Further, if then we have

(17) | ||||

We call the values appearing in
the nine cases from (13) to (17) *irregular* values.
Otherwise is called a *regular* value.

Step (ii). Taking and in Lemma 2.5 we get

(18) | ||||

We see clearly that we can assume either (ii.1): or (ii.2): and . Moreover, by taking in (18) we see that in (16) can be expressed by -linear combinations of MZVs appeared in the (13) and (14). The argument is similar for in (16). Therefore in what follows we assume that are always regular and show that they can be expressed by -linear combinations of MZVs of the same weight and depth three or less.

Step (ii.1). Let . Then we must have either or since we assume

is regular. Then we may use and in Lemma 2.5 to get

By symmetry we only need to consider

(19) |

where . But now we may take and in Lemma 2.5 to reduce it to either or . Then we get the following two kind of values:

where . But is expressible by MZVs of the same weight and of depth three by Prop. 2.4. Case (ii.1) is proved.

Step (ii.2). Let and . Consider

To guarantee convergence we must have , , , , and . Taking and in Lemma 2.5 we get

(20) | ||||

(21) |

So we may assume that either (ii.2.1): or (ii.2.2): and , or (ii.2.3): and . Here (ii.2.1) comes from (20) while (ii.2.2) and (ii.2.3) come from (21).

Step (ii.2.1). Let . Then we must have either or in

since we consider only regular values only. Thus we may put and in Lemma 2.5 to get

It is easy to see that all the triple zeta values above have the same weight. We remind the reader that to determine the weight of a MZV it is not enough just to add up all the components. One also needs to check that every component is positive.

Step (ii.2.2). Let and . By (12), to guarantee convergence of

we need to assume

(22) |

Moreover, since is regular we must have . Putting , and in Lemma 2.5 we get

(23) | ||||

We point out that since we have used we need to check the convergence of all the above three kinds of values under the assumption (22) even though the checking itself is trivial.

Returning to the reduction of (23) we see that the last two sums are expressible by MZVs so we only need to consider those values appearing in the first sum, namely, those of the form in the next case.

Step (ii.2.3). Let and . Then we must have since we only consider regular values. To guarantee convergence of