ANALYTIC CONTINUATION OF MULTIPLE ZETA-FUNCTIONS AND THEIR VALUES AT NON-POSITIVE INTEGERS

SHIGEKI AKIYAMA, SHIGEKI EGAMI AND YOSHIO TANIGAWA

Abstract. Analytic continuation of the multiple zeta-function is established by a simple application of the Euler-Maclaurin summation formula. Multiple zeta values at non-positive integers are defined and their properties are investigated.

1. Introduction

The multiple zeta values due to D. Zagier are defined by

ζ_{k}(s_{1}, s_{2}, …, s_{k}) = ∑_{0<n_{1}<n_{2}<...<n_{k}} \frac{1}{n_{1}^{s_{1}}n_{2}^{s_{2}} … n_{k}^{s_{k}}}

with positive integers s_{i} (i = 1, 2, …, k) and s_{k} ≥ 2. These values have a certain connection with topology and physics, and algebraic relations among them are extensively studied (see [18], [19], [6], [7] and [14]). Recently, Y. Ohno developed a unified algebraic relation in [16]. It is also interesting to consider it for complex variables s_{i}.

In this paper, we treat analytic continuation of ζ_{k}(s_{1}, s_{2}, …, s_{k}). Analytic continuation of ζ_{2}(s_{1}, s_{2}) was proved by F.V. Atkinson [5] with applications to the study of the asymptotic behavior of the ‘mean values’ of zeta-functions. See also Y. Motohashi [15] and M. Katsurada & K. Matsumoto [13]. In [4], T. Arakawa & M. Kaneko used analytic continuation of ζ_{k}(s_{1}, s_{2}, …, s_{k}) as a function of one variable s_{k} when s_{1}, s_{2}, …, s_{k−1} are positive integers, and discussed the relation among generalized Bernoulli numbers. On the other hand, S. Egami discussed the relationship among various multiple zeta-functions introduced by E.W. Barnes, T. Shintani and D. Zagier. (See [9] and [10].)

However, for a general k, we cannot find the proof of analytic continuation of ζ_{k}(s_{1}, s_{2}, …, s_{k}) as a function of k variables in literature (but see the comment of Zagier [18, p. 509, lines 14–19]). We shall show that the multiple zeta-function can be continued analytically to ℂ^{k} and discuss interesting properties of multiple zeta values at non-positive integers.

1991 Mathematics Subject Classification. Primary 11M41; Secondary 30B40,32Dxx.

The authors wish to express their gratitude to the referee for valuable comments on the earlier version of the present paper. The third author also thanks Professor Aleksandar Ivić for giving him useful comments. Remark 1. After submitting the first version of our paper, we found a recent work of J. Zhao [20] treating analytic continuation of multiple zetafunction. This fact was also pointed out by the referee. With the help of the theory of generalized function in the sense of I.M. Gel’fand and G.E. Shilov, he gave their possible singularities as well as the residues. However our method is apparently simple and reveals the exact location of singularities, which seems to be an advantage.

2. Analytic continuation

Let l and m be positive integers. Define an entire function:

ϕ_{l}(m, s) = ∑_{n=1}^{m} \frac{1}{n^{s}} − \{\frac{m^{1−s} − 1}{1 − s} + \frac{1}{2m^{s}} − ∑_{q=1}^{l} \frac{(s)_{q}a_{q}}{m^{s+q}} + ζ(s) − \frac{1}{s − 1}\} (1)

with (s)_{n} = s(s + 1) ⋯ (s + n − 1) and a_{q} = B_{q+1}/(q + 1)!. Here B_{q} are Bernoulli numbers defined by z/(e^{z} −1) = ∑_{q=0}^{∞} B_{q}z^{q}/q! and ζ(s) is the Riemann zeta-function. By using the Euler-Maclaurin summation formula, we have ϕ_{l}(m, s) = O(|(s)_{l+1}|m^{−ℜ(s)−l−1}) when s is a complex number. Considering s as a complex variable and m → ∞, we get an analytic continuation of ζ(s) in ℜ(s + l + 1) > 0. Note that (1) is also valid when s → 1, if we replace (m^{1−s} − 1)/(1 − s) by its limit log m.

This is one of the oldest way of the analytic continuation of the Riemann zeta-function, which provides us with a method of numerical calculations in the critical strip 0 < ℜs < 1. (c.f. [8], [12]). It does not give us the celebrated functional equation of ζ(s) directly, but it is possible to derive it by more precise observations (see Chapter 2 of [17]). Hereafter we will use (1) in the form:

∑_{n=m+1}^{∞} \frac{1}{n^{s}} = −ϕ_{l}(m, s) + \frac{m^{1−s}}{s − 1} − \frac{1}{2m^{s}} + ∑_{q=1}^{l} \frac{(s)_{q}a_{q}}{m^{s+q}} , (2)

for ℜ(s) > 1. Consider the multiple zeta-function in two variables:

ζ_{2}(s_{1}, s_{2}) = ∑_{0<n_{1}<n_{2}} \frac{1}{n_{1}^{s_{1}}n_{2}^{s_{2}}}

ANALYTIC CONTINUATION OF MULTIPLEZETA-FUNCTIONS AND THEIR VALUESAT NON-POSITIVE INTEGERS3

with ℜs_{i} > 1 (i = 1, 2). By (2),

ζ_{2}(s_{1}, s_{2}) = ∑_{n_{1}=1}^{∞} \frac{1}{n_{1}^{s_{1}}} ∑_{n_{2}=n_{1}+1}^{∞} \frac{1}{n_{2}^{s_{2}}}

= ∑_{n_{1}=1}^{∞} \frac{1}{n_{1}^{s_{1}}} \{−ϕ_{l}(n_{1}, s_{2}) + \frac{n_{1}^{1−s_{2}}}{s_{2} − 1} − \frac{1}{2n_{1}^{s_{2}}} + ∑_{q=1}^{l} \frac{(s_{2})_{q}a_{q}}{n_{1}^{s_{2}+q}} \}

= \frac{ζ(s_{1} + s_{2} − 1)}{s_{2} − 1} − \frac{ζ(s_{1} + s_{2})}{2}

+ ∑_{q=1}^{l}(s_{2})_{q}a_{q}ζ(s_{1} + s_{2} + q) − ∑_{n_{1}=1}^{∞} \frac{ϕ_{l}(n_{1}, s_{2})}{n_{1}^{s_{1}}} (3)

holds for ℜ(s_{i}) > 1 (i = 1, 2). The terms on the right hand side have meromorphic continuations except the last one. The last sum is absolutely convergent, and hence holomorphic, in ℜ(s_{1} +s_{2} +l) > 0. Thus we now have a meromorphic continuation of ζ_{2}(s_{1}, s_{2}) to ℜ(s_{1} + s_{2} + l) > 0. Since we can choose arbitrary large l, we get a meromorphic continuation of ζ_{2}(s_{1}, s_{2}) to ℂ^{2}, which is holomorphic in

{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{2} ≠ 1, s_{1} + s_{2} ∉ {2, 1, 0, −2, −4, −6, … }} .

One can see easily that this trick can be applied to a multiple zeta-function with k variables. In fact,

ζ_{k}(s_{1}, s_{2}, …, s_{k}) = ∑_{n_{1}=1}^{∞} \frac{1}{n_{1}^{s_{1}}} ∑_{n_{2}=n_{1}+1}^{∞} \frac{1}{n_{2}^{s_{2}}} … ∑_{n_{k−1}=n_{k−2}+1}^{∞} \frac{1}{n_{k−1}^{s_{k−1}}} ∑_{n_{k}=n_{k−1}+1}^{∞} \frac{1}{n_{k}^{s_{k}}}

= ∑_{n_{1}=1}^{∞} \frac{1}{n_{1}^{s_{1}}} ∑_{n_{2}=n_{1}+1}^{∞} \frac{1}{n_{2}^{s_{2}}} … ∑_{n_{k−1}=n_{k−2}+1}^{∞} \frac{1}{n_{k−1}^{s_{k−1}}} ×

× \{−ϕ_{l}(n_{k−1}, s_{k}) + \frac{n_{k−1}^{1−s_{k}}}{s_{k} − 1} − \frac{1}{2n_{k−1}^{s_{k}}} + ∑_{q=1}^{l} \frac{(s_{k})_{q}a_{q}}{n_{k−1}^{s_{k}+q}} \}

= \frac{ζ_{k−1}(s_{1}, s_{2}, …, s_{k−2}, s_{k−1} + s_{k} − 1)}{s_{k} − 1} − \frac{ζ_{k−1}(s_{1}, s_{2}, …, s_{k−2}, s_{k−1} + s_{k})}{2}

+ ∑_{q=1}^{l}(s_{k})_{q}a_{q}ζ_{k−1}(s_{1}, s_{2}, …, s_{k−2}, s_{k−1} + s_{k} + q)

(4) − ∑_{0<n_{1}<n_{2}<...<n_{k−1}} \frac{ϕ_{l}(n_{k−1}, s_{k})}{n_{1}^{s_{1}}n_{2}^{s_{2}} … n_{k−1}^{s_{k−1}}}

for ℜ(s_{i}) > 1 (i = 1, 2, …, k). Since

∑_{0<n_{1}<n_{2}<...<n_{k−1}} \frac{ϕ_{l}(n_{k−1}, s_{k})}{n_{1}^{s_{1}}n_{2}^{s_{2}} … n_{k−1}^{s_{k−1}}} ≪ ∑_{n_{k−1}} \frac{n_{k−1}^{−l−ℜ(s_{k})+k−3}}{n_{k−1}^{L}}

with L = ℜ(s_{k−1}) + ∑_{{1≤j≤k−2, \atop ℜ(s_{i})≤0}} ℜ(s_{i}), the last summation is convergent abso-

lutely in

l − k + 2 + ℜ(s_{k}) + ℜ(s_{k−1}) + ∑_{{1≤i≤k−2, \atop ℜ(s_{i})≤0}} ℜ(s_{i}) > 0. (5)

Since l can be taken arbitrarily large, we get an analytic continuation of ζ_{k}(s_{1}, s_{2}, …, s_{k}) to ℂ^{k}. Now we consider the set of singularities. For simplicity, we put (s)_{0} = 1. Then the ‘singular part’ of ζ_{2}(s_{1}, s_{2}) is given by

\frac{ζ(s_{1} + s_{2} − 1)}{s_{2} − 1} + ∑_{q_{1}≥0} \frac{a_{q_{1}}(s_{2})_{q_{1}}}{s_{1} + s_{2} + q_{1} − 1}.

Note that this sum is formal and only indicates local singularities. From this expression, we see

s_{2} = 1, s_{1} + s_{2} ∈ {2, 1, 0, −2, −4, −6, … }

forms the set of whole singularities. For the case ζ_{3}(s_{1}, s_{2}, s_{3}), by using the singular part of ζ_{2}, we see that singularities lie on

s_{3} = 1, s_{2} + s_{3} ∈ {2, 1, 0, −2, −4, −6, … }

and

s_{1} + s_{2} + s_{3} ∈ {3, 2, 1, 0, −1, −2, −3, … }.

We want to show that these are the whole singularities. It suffices to show that no singularities defined by one of above equations will identically vanish. This can be shown by replacing variables:

u_{1} = s_{1}, u_{2} = s_{2} + s_{3}, u_{3} = s_{3}.

In fact, we see that the singular part of ζ_{3}(u_{1}, u_{2} − u_{3}, u_{3}) is given by

\frac{1}{u_{3} − 1}ζ_{2}(u_{1}, u_{2} − 1) + ∑_{q_{2}≥0}(u_{3})_{q_{2}}a_{q_{2}}ζ_{2}(u_{1}, u_{2} + q_{2}).

By this expression we see that the singularities of ζ_{2}(u_{1}, u_{2} + q) are summed with functions of u_{3} of different degree. Thus these singularities, as a weighted sum by another variable u_{3}, will not vanish identically. Similarly, we see Theorem 1. The multiple zeta-function ζ_{k}(s_{1}, s_{2}, …, s_{k}) is meromorphically continued to ℂ^{k} and has singularities on

s_{k} = 1, s_{k−1} + s_{k} = 2, 1, 0, −2, −4, …,

and

∑_{i=1}^{j} s_{k−i+1} ∈ ℤ_{≤j} (j = 3, 4, …, k),

where ℤ_{≤j} is the set of integers less than or equal to j.

3. Zeta values at non-positive integers

In this section, we use the notation (s)_{0} = 1 and (s)_{−1} = 1/(s − 1) for the sake of simplicity. We also put a_{q} = B_{q+1}/(q + 1)! for q = 0 and −1 as in §2. A point of ℂ^{n} (n ≥ 2) is said to be a point of indeterminacy of a meromorphic function if both the local denominator and the local numerator vanish there. See p.164 of [11] for the precise definition. For instance, let f(s_{1}, s_{2}) = s_{1}/(s_{1} + s_{2}). Then s_{1} = s_{2} = 0 is a point of indeterminacy of f. So the value of f at (0, 0) depends on a limiting process, for example lim_{s_{2}→0} lim_{s_{1}→0} f(s_{1}, s_{2}) = 0 while lim_{s_{1}→0} lim_{s_{2}→0} f(s_{1}, s_{2}) = 1.

Let r_{i} (i = 1, 2, …, k) be non-negative integers. Recall from Theorem 1 that each point (−r_{1}, −r_{2}, …, −r_{k}) except when k = 2 and r_{1} + r_{2} is odd, lies on the set of singularities. Moreover, such a point is a point of indeterminacy. To prove this, it suffices to show that ζ_{k} has a finite value at (−r_{1}, −r_{2}, …, −r_{k}) by a specific limiting process. Now we give the definition which we will employ in this paper. Definition . We define the multiple zeta values at non-positive integers by

ζ_{k}(−r_{1}, −r_{2}, …, −r_{k}) = lim_{s_{1}→−r_{1}} lim_{s_{2}→−r_{2}} ⋯ lim_{s_{k}→−r_{k}} ζ_{k}(s_{1}, s_{1}, …, s_{k}).

From (4) and the above definition, we have

ζ_{k}(−r_{1}, −r_{2}, …, −r_{k})

= ∑_{q=−1}^{r_{k}}(−r_{k})_{q}a_{q}ζ_{k−1}(−r_{1}, −r_{2}, …, −r_{k−2}, −r_{k−1} − r_{k} + q). (6)

Here we used the fact that ϕ_{r}(m, l) = 0 for l ≥ r. This formula shows that the value ζ_{k}(−r_{1}, −r_{2}, …, −r_{k}) is determined recursively as a finite number, hence each point (−r_{1}, −r_{2}, …, −r_{k}) is a point of indeterminacy. The formula (6) also gives us a simple way of calculation of multiple zeta values ζ_{k}(−r_{1}, −r_{2}, …, −r_{k}). For example, we have ζ_{2}(0, 0) = 1/3, ζ_{3}(0, 0, 0) = −1/4, ζ_{4}(0, 0, 0, 0) = 1/5, ζ_{2}(−1, −1) = 1/360, ζ_{3}(−1, −1, −1) = 83/30240. One may expect that ζ_{k}(0, 0, …, 0) = (−1)^{k}/(1 + k). This assertion will be proved in the forthcoming paper [2]. Here we shall show some other interesting properties.

Theorem 2. Let r_{i} (i = 1, 2, …, k) be non-negative integers. Then the value ζ_{k}(−r_{1}, −r_{2}, …, −r_{k}) is a rational number whose denominator has prime factors less than or equal to 1 + k + ∑_{i=1}^{k} r_{i}.

Proof. It is well known that ζ(0) = −1/2, ζ(−2r) = 0 and ζ(1 − 2r) = −B_{2r}/2r for positive integers r. By using the theorem of von Staudt & Clausen, the assertion for k = 1 is obvious. From (6), the proof is completed by the induction on k.

Theorem 3. Let r_{i} (i = 1, 2, …, k) be positive integers and n_{i} (i = 1, 2, …, k) be non-negative integers. If ∑_{i=1}^{k}(r_{i} + n_{i} + 1) is odd then

∑_{σ∈𝔖_{k}} sgn(σ)ζ_{k}(−r_{σ(1)} − n_{1}, −r_{σ(2)} − n_{2}, …, −r_{σ(k)} − n_{k}) = 0, (7)

where 𝔖_{k} is the symmetric group of degree k and sgn(σ) is the signature of σ ∈ 𝔖_{k}

The statement is trivial when r_{i} are not distinct. We will show some examples when n_{1} = n_{2} = n_{3} = 0 before proving the theorem:

ζ_{2}(−1, −2) − ζ_{2}(−2, −1) = −\frac{1}{240} + \frac{1}{240} = 0,

ζ_{3}(−1, −2, −3) + ζ_{3}(−2, −3, −1) + ζ_{3}(−3, −1, −2)

− ζ_{3}(−1, −3, −2) − ζ_{3}(−2, −1, −3) − ζ_{3}(−3, −2, −1)

= −\frac{101}{100800} + \frac{149}{302400} + \frac{107}{302400} + \frac{19}{30240} + \frac{17}{43200} − \frac{131}{151200} = 0.

Proof. In the following, we shall only prove the case n_{1} = n_{2} = … = n_{k} = 0. The generalization is quite easy and is left to the reader. Let

I_{k} = {(−r_{1}, …, −r_{k})|r_{i} are positive integers and ∑_{i=1}^{k}(r_{i} + 1) is odd.} .

For 1 ≤ a < b ≤ k, we define a vector space 𝔉_{k}(a, b) consisting of ℂ-valued functions f(ξ_{1}, ξ_{2}, …, ξ_{k}) such that

∑_{σ∈𝔖_{a:b}} sgn(σ)f(−r_{σ(1)}, −r_{σ(2)}, …, −r_{σ(k)}) = 0

for any (−r_{1}, …, −r_{k}) ∈ I_{k}, where 𝔖_{a:b} is a subgroup of 𝔖_{k} whose elements stabilize {1, 2, …, k} \ {a, a + 1, …, b}. Our task is to show that the function ζ_{k}(ξ_{1}, ξ_{2}, …, ξ_{k}) is contained in 𝔉_{k}(1, k). Considering the coset decomposition 𝔖_{k}/𝔖_{a:b}, we see 𝔉_{k}(a, b) is a subspace of 𝔉_{k}(1, k). Thus it is enough to show that the multiple zeta-function is contained in a sum of subspaces 𝔉_{k}(a, b).

First we prove the case k ≤ 3. The assertion (7) is valid when k = 1, since ζ(−2r) = 0 for any positive integers r. When k = 2 and (−r_{1}, −r_{2}) ∈ I_{2}, we have

ζ_{2}(−r_{1}, −r_{2}) = ∑_{q=−1}^{r_{2}}(−r_{2})_{q}a_{q}ζ(−r_{1} − r_{2} + q)

= −\frac{1}{2}ζ(−r_{1} − r_{2}), (8)

which shows the assertion for k = 2. When k = 3 and (−r_{1}, −r_{2}, −r_{3}) ∈ I_{3}, we have from (8) that

ζ_{3}(−r_{1}, −r_{2}, −r_{3}) = ∑_{q=−1}^{r_{3}}(−r_{3})_{q}a_{q}ζ_{2}(−r_{1}, −r_{2} − r_{3} + q)

= −\frac{1}{2}ζ_{2}(−r_{1}, −r_{2} − r_{3}) − \frac{1}{2} ∑_{q : odd^{q=−1}}^{r_{3}}(−r_{3})_{q}a_{q}ζ(−r_{1} − r_{2} − r_{3} + q). (9)

Hence ζ_{3}(ξ_{1}, ξ_{2}, ξ_{3}) ∈ 𝔉_{3}(2, 3) + 𝔉_{3}(1, 2).

Let k ≥ 3 and (−r_{1}, …, −r_{k}) ∈ I_{k}. Then by induction on k, we can easily see that

ζ_{k}(ξ_{1}, ξ_{2}, …, ξ_{k}) + \frac{1}{2}ζ_{k−1}(ξ_{1}, ξ_{2}, …, ξ_{k−2}, ξ_{k−1} + ξ_{k})

∈ 𝔉_{k}(1, 2) + 𝔉_{k}(2, 3) + … + 𝔉_{k}(k − 2, k − 1). (10)

The assertion of the theorem follows immediately from (10).

Suppose that k = 3, r_{i} are non-negative integers, r_{1} > 0 and r_{1} + r_{2} + r_{3} is even. Then from (8) and (9), we have

ζ_{3}(−r_{1}, −r_{2}, −r_{3}) = −\frac{1}{2}\{ζ_{2}(−r_{1} − r_{2}, −r_{3}) + ζ_{2}(−r_{1}, −r_{2} − r_{3})\}. (11)

One may expect that symmetric expressions like (8) and (11) would give us a deeper understanding of Theorem 3. Further calculation suggests us the following conjecture. To state it, we shall prepare some notation. Let S be the ordered index set {1, 2, ⋯, k} of k elements and let D_{l}^{k} be the set of all ways of dividing S into l parts. Clearly the set D_{l}^{k} consists of \binom{k−1}{l−1} elements. The element J in D_{l}^{k} can be expressed as

J = (1, ⋯, i_{1}| i_{1} + 1, ⋯, i_{2}| i_{2} + 1, ⋯, i_{l−1}| i_{l−1} + 1, ⋯, k).

Let A = (−r_{1}, −r_{2}, ⋯, −r_{k}) be a sequence of k non-positive integers. For J ∈ D_{l}^{k} as above, we set

A^{J} = (−r_{1} − r_{2} − ⋯ − r_{i_{1}}, −r_{i_{1}+1} − ⋯ − r_{i_{2}}, ⋯, −r_{i_{l−1}+1} − ⋯ − r_{k})

and

ζ_{l}(A^{J}) = ζ_{l}(−r_{1} − r_{2} − ⋯ − r_{i_{1}}, −r_{i_{1}+1} − ⋯ − r_{i_{2}}, ⋯, −r_{i_{l−1}+1} − ⋯ − r_{k}).

Now we can state our

Conjecture . Let r_{i} be non-negative integers, r_{1} > 0 and ∑_{i=1}^{k}(r_{i} + 1) is odd. Let A = (−r_{1}, −r_{2}, ⋯, −r_{k}). Then we have

ζ_{k}(A) = −2 ∑_{j=1}^{k−1}(2^{j+1} − 1)\frac{B_{j+1}}{j + 1}( ∑_{J∈D_{k−j}^{k}} ζ_{k−j}(A^{J})). (12)

Further discussion ^{1} will be found in [2]. We would like to note that we could find the conjectural form of (12) by the home page ‘Sloane’s On-Line Encyclopedia of Integer Sequences’.

Theorem 4. For a positive integer r, we have

\frac{ζ(−4r − 1)}{ζ_{2}(−2r, −2r)} = (2r + 1)\binom{4r + 2}{2r + 1}.

Proof. From (6) and the definition of a_{q}, we have

ζ_{2}(−2r, −2r) = \frac{B_{4r+2}}{2(2r + 1)^{2}} + \frac{1}{2r + 1} ∑_{j=1}^{r} \binom{2r + 1}{2j}\frac{B_{2j}B_{4r+2−2j}}{4r + 2 − 2j} .

We note the following identity of Bernoulli numbers:

2(2r + 1) ∑_{j=0}^{r} \binom{2r + 1}{2j}\frac{B_{2j}B_{4r+2−2j}}{4r + 2 − 2j} + \frac{((2r + 1)!)^{2}}{(4r + 2)!} B_{4r+2} = 0,

obtained by putting m = n = 2r + 1 and x = 0 in Apostol [3, p.276, 19 (b)]. Hence,we have

ζ_{2}(−2r, −2r) = −\frac{1}{2(2r + 1)^{2}} \frac{((2r + 1)!)^{2}}{(4r + 2)!} B_{4r+2}.

On the other hand, ζ(−4r − 1) = −B_{4r+2}/(4r + 2), and this gives the assertion of Theorem 4.

Finally we want to add several remarks.

1Addendum for the revised version: Finally we have succeeded in proving the validity of this Conjecture. See [2] for details.

Remark 2. There are many other possibilities for the definition of multiple zeta values at non-positive integers. For instance, define the value ζ_{k}^{∗} by

ζ_{k}^{∗}(−r_{1}, −r_{2}, …, −r_{k}) = lim_{ε→0} ζ_{k}(−r_{1} + ε, −r_{2} + ε, …, −r_{k} + ε).

When k = 2, this is equivalent to define by

ζ_{2}^{∗}(−r_{1}, −r_{2}) = ∑_{q=−1}^{r_{1}}(−r_{2})_{q}a_{q}ζ(−r_{1} − r_{2} + q) + \frac{(−1)^{r_{1}}r_{1}! r_{2}! a_{r_{1}+r_{2}+1}}{2} (13)

for non-negative integers r_{i} (i = 1, 2). This definition seems to be better than our former definition at least when k = 2, 3. In fact, when r_{1} + r_{2} is odd we have ζ_{2}(−r_{1}, −r_{2}) = ζ_{2}^{∗}(−r_{1}, −r_{2}) and

ζ_{2}^{∗}(−2u_{1}, −2u_{2}) + ζ_{2}^{∗}(−2u_{2}, −2u_{1}) = 0

for positive integers u_{i} (i = 1, 2). Especially we have ζ_{2}^{∗}(−2u, −2u) = 0 with a positive integer u. We can also find a recursive formula for k = 3 and show that

ζ_{3}^{∗}(−2u, −2v, −2w) + ζ_{3}^{∗}(−2w, −2v, −2u) = 0

for positive integers u, v, w. However in the general case, it seems difficult to construct a recursive formula like (13), since there exist a lot of singularities to take into account. One may hope that

ζ_{k}^{∗}(−2u, −2u, …, −2u) = 0

for a positive integer u.

Remark 3. The set of points of indeterminacy forms a k − 2 dimensional holomorphic subvariety of ℂ^{k}, by p.166 of [11]. We have shown that each non-positive points (−r_{1}, …, −r_{k}) are actually on this subvariety, but there are another type of integer points on this set. For instance, it will be shown in [2] that (−r_{1}, …, −r_{k−1}, 1) is a point of indeterminacy whose multiple zeta value in our sense is rational, when r_{i} ∈ ℤ_{≥0} and not all r_{i} is zero. Also we have

ζ_{3}(4, −3, −2) = −\frac{461}{2520} − \frac{π^{2}}{144} + \frac{π^{4}}{45360} + \frac{ζ(3)}{420}.

We could not determined yet the whole set of such integer points.

Remark 4. We can easily apply the Euler-Maclaurin summation formula to more general zeta-functions. For instance, let α_{i} > −1 (i = 1, 2, …, k) be real numbers and χ_{i} (i = 1, 2, …, k) the Dirichlet characters. Define a function ξ(s_{1}, s_{2}, …, s_{k}) for ℜ(s_{i}) > 1 (i = 1, 2, …, k), by a convergent sum:

∑_{0<n_{1}<n_{2}<...<n_{k}} \frac{χ_{1}(n_{1})χ_{2}(n_{2}) ⋯ χ_{k}(n_{k})}{(n_{1} + α_{1})^{s_{1}}(n_{2} + α_{2})^{s_{2}} … (n_{k} + α_{k})^{s_{k}}} .

Then ξ is meromorphically continued to ℂ^{k}. In fact, using binomial series expansion of (n + β)^{−s} = n^{−s}(1 + β/n)^{−s} for each variable, we see that ξ can be expressed in terms of absolutely convergent sums of multiple zeta functions. See [1] for further study of this kind of function.

References

[1] S. Akiyama and H. Ishikawa, An analytic continuation of multiple L-functions and related zeta-functions, to appear in ‘Analytic Number Theory’ ed. by C.Jia and K.Matsumoto.

[2] S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, submitted.

[3] T.M. Apostol, Introduction to Analytic Number Theory, Springer 1976.

[4] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta-functions, Nagoya Math. J. 153 (1999), 189–209.

[5] F.V. Atkinson, The mean value of the Riemann zeta-function, Acta Math., 81 (1949), 353–376.

[6] J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisoněk, Combinatorial aspects of multiple zeta values, Electron. J. Combin., 5 (1998), no. 1, Research Paper 38, 12 pp.

[7] D.J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett., B 393 (1997), no. 3-4, 403–412.

[8] H.M. Edwards,Riemann’s Zeta Function, Academic Press, New York and London 1974.

[9] S. Egami, Reciprocity laws of multiple zeta functions and generalized Dedekind sums, Analytic number theory and related topics (Tokyo, 1991), 17–27, World Sci. Publishing, River Edge, NJ, 1993.

[10] S. Egami, Introduction to multiple zeta function, Lecture Note at Niigata University (in Japanese), DVI and TeX files are available at http://mathalg.ge.niigatau.ac.jp/Seminar/Intensive/Egami.html

[11] R.C. Gunning, Introduction to holomorphic functions of several variables II, Wadsworth & Brooks/Cole Mathematics Series.

[12] A. Ivić, The Riemann Zeta-Function, A Wiley-Interscience Publ. John Wiley & Sons 1985.

[13] M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet L-functions. Math. Z., 208 (1991), 23–39.

[14] T.Q.T. Le and J. Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl., 62 (1995), 193– 206.

[15] Y. Motohashi, A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad., Ser. A Math. Sci. 61 (1985), 222–224.

[16] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74 (1999), 39–43.

[17] E.C. Titchmarsh (revised by D.R. Heath-Brown), The theory of the Riemann Zetafunction, Clarendon Press, Oxford 1986.

[18] D. Zagier, Values of zeta-functions and their applications, First European Congress of Mathematics, Vol. II, Birkhäuser, (1994) 210–220 [19] D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta

values, Research into automorphic forms and L functions (in Japanese) ( Kyoto, 1992), Sūrikaisekikenkyūsho Kōkyūroku, 843 (1993), 162–170.

[20] J. Zhao, Analytic continuation of multiple zeta function, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1275–1283.

Shigeki AKIYAMA Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050,Niigata 950-2181, Japan e-mail: akiyama@mathalg.ge.niigata-u.ac.jp

Shigeki EGAMI Department of Mechanical and Intelligent Systems Engineering, Faculty of Engineering, Toyama University Gofuku 3190, Toyama 930-8555,Japan e-mail: megami@eng.toyama-u.ac.jp

Yoshio TANIGAWA Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya 464-8602, Japan e-mail: tanigawa@math.nagoya-u.ac.jp