Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points

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by  Kohji Matsumoto

LEMZ12.tex January 25, 2016
LAURENT SERIES EXPANSIONS OF MULTIPLE
ZETA-FUNCTIONS OF EULER-ZAGIER TYPE AT INTEGER
POINTS
KOHJI MATSUMOTO, TOMOKAZU ONOZUKA, AND ISAO WAKABAYASHI
Abstract. Wegiveexplicitexpressions(oratleastanalgorithmofobtaining
such expressions) of the coefficients of the Laurent series expansions of the
6 Euler-Zagiermultiple zeta-functions at any integer points. The main tools are
1 the Mellin-Barnes integral formula and the harmonic product formulas. The
0
Mellin-Barnes integralformula is used in the induction process on the number
2
ofvariables,andtheharmonicproductformulaisusedtoshowthattheLaurent

n
series expansion outside the domain of convergencecanbe obtained from that
a
inside the domain of convergence.
J

2
2 1. Introduction and the statement of main results

] The Euler-Zagier r-ple zeta-function is defined by ([5], [15])
T
N ∞ ∞ ∞
(1.1) ζ (s) = ··· n−s1(n +n )−s2···(n +···+n )−sr
h. r 1 1 2 1 r
t nX1=1nX2=1 nXr=1
a
m (where s = (s1,...,sr) ∈ Cr) in the domain of its absolute convergence, which is
[ (1.2) D = {s ∈ Cr | ℜ(s(j,r)) > r −j +1 (1 ≤ j ≤ r)},
r

1
where s(j,r) = s + s + ··· + s ([7, Theorem 3]). Special values ζ (m)
v j j+1 r r
8 (m = (m1,...,mr) ∈ Zr) of (1.1) in this domain are called multiple zeta values
1 (MZV), and have been studied extensively.
9 It is known that (1.1) can be continued meromorphically to the whole space Cr
5
0 (Akiyama, Egami and Tanigawa [1], Zhao [16]). Therefore the behavior of (1.1)
1. around the points m ∈ Zr outside the domain D is also of great interest. This
r
0 direction of research was initiated by Akiyama, Egami and Tanigawa [1], and
6
then pursued further by subsequent mathematicians (Akiyama and Tanigawa [2],
1
: Komori [6], Sasaki [13] [14], and the second-named author [11]).
v
i In this paper, to understand the behavior of (1.1) around the integer points
X
more closely, we give Laurent series expansions for the Euler-Zagier r-ple zeta-
r
a function at integer points. When r = 1, the function ζ (s) is nothing but the
1
Riemann zeta-function, and this function has, at m ∈ Z, the following Taylor or
Laurent series expansion:
∞
1
ζ(n)(m)(s−m)n (m 6= 1),
n!
(1.3) ζ(s) = n=0
X ∞
 1 + γn(s−1)n (m = 1),
s−1
n=0
 X


This researchwas partially supported by Grants-in-Aidfor Scientific Research,GrantNum-
bers25287002(forthefirst-namedauthor)and13J00312(forthesecond-namedauthor),JSPS.
1
2 KOHJIMATSUMOTO, TOMOKAZU ONOZUKA,AND ISAOWAKABAYASHI
where γ is the n-thStieltjes constant (or generalized Euler constant)† andζ(n)(s)
n
is the n-th derivative of the Riemann zeta-function. The 0-th Stieltjes constant
γ = γ isthewell-knownEuler constant, andStieltjesconstantshavebeenstudied
0
by several authors; see [3, p.164]. Note that for m > 1, ζ(n)(m) is given by the
series ∞ (−logk)n/km. (In this paper, we define (−log1)0 = 00 = 1.)
k=1
The same type of expansions holds in the multivariable case. Generally, for
a funcPtion f(s) and (n ,...,n ) ∈ Z , we denote by f(n1,...,nr)(s) the (partial)
1 r ≥0
derivative (∂n1/∂sn1)···(∂nr/∂snr)f(s). When m ∈ D , obviously
1 r r
(1.4)
∞ ∞
1
ζ (s) = ··· ζ(n1,...,nr)(m ,...,m )(s −m )n1···(s −m )nr
r n !···n ! r 1 r 1 1 r r
1 r
nX1=0 nXr=0
when s is close to m.
Also, we define the r-ple (n ,...,n )-th Stieltjes constant γ by the fol-
1 r (n1,...,nr)
lowing formula which is valid when s is close to (1,...,1):
r
1 1
(1.5) ζ (s ,...,s ) =
r 1 r
s +···+s −(r −k +1) s +···+s −r
k=2 k r !(cid:26) 1 r
Y
∞ ∞
+ ··· γ (s −1)n1···(s −1)nr .
(n1,...,nr) 1 r
)
nX1=0 nXr=0
When r = 1, (regarding the empty product as 1) we find that (1.5) is reduced to
the second formula of (1.3), so γ = γ .
(n1) n1
In Section 3, we will prove the above (1.5), which we call the Laurent series
expansion at (s ,...,s ) = (1,...,1) of the Euler-Zagier r-ple zeta-function (see
1 r
Lemma 3.1).
Remark 1.1. In the present paper we use the term ”Laurent series expansion”
in the above extended way, that is, the linear factors in the denominator and in
the numerator are not necessarily the same. The standard definition of Laurent
series expansion is more restricted; see [4, Chapter I, Section 5, pp.88-90] or [12,
Erstes Kapitel, Sektion 20, pp.68-70].
Remark 1.2. It is not clear what is the most suitable definition of multiple
Stieltjes constants. See Remark 4.4 at the end of Section 4.
The first main theorem in this paper is as follows.
Theorem 1.3. Let r,m ,...,m be positive integers, and put m = (m ,...,m ).
1 r 1 r
Then the coefficients in the Laurent series of ζ (s) at s = m can be given explicitly
r
byusing γ andζ(l1,...,lk)(q ,...,q ) where1 ≤ k ≤ r, (n ,...,n ),(l ,...,l ) ∈
(n1,...,nk) k 1 k 1 k 1 k
(Z )k and (q ,...,q ) ∈ D ∩(Z )k.
≥0 1 k k ≥1
Here we give some comments on the meaning of this theorem.
(i) When r = 1, this theorem is trivial, because the assertion is nothing but
(1.3) for m ≥ 1. Similarly, in the general r-ple case for r ≥ 1, the theorem is
trivial by (1.4) when m ∈ D ∩ (Z )r, and follows directly from (1.5) when
r ≥1
†Some authors call (−1)nn!γn the n-th Stieltjes constant.
LAURENT SERIES EXPANSIONS OF MULTIPLE ZETA-FUNCTIONS 3
m = (1,...,1). The main point of the theorem lies in the cases when m ∈
(Z )r \(D ∪(1,...,1)), which happens only when r ≥ 2.
≥1 r
(ii) When m ∈ (Z )r \ (D ∪ (1,...,1)), we understand the meaning of the
≥1 r
”Laurent series” at s = m in the extended sense mentioned in Remark 1.1. More
strictly, when m > 1 and m = ··· = m = 1 (1 ≤ j ≤ r−1), then the Laurent
j j+1 r
series at s = m is of the form of a fraction, whose denominator is the product
of linear factors (s(k,r) − (r −k + 1)) (j + 1 ≤ k ≤ r), and its numerator is a
Taylor series with respect to (s −m ),...,(s −m ). An example will be given
1 1 r r
in Example 3.3 at the end of Section 3. The term ”coefficients in the Laurent
series” inthestatement ofthetheoremmeans thecoefficients inthisTaylor series.
As can be seen in Example 3.3, our proof of Theorem 1.3 gives an algorithm of
obtaining the coefficients explicitly.
After mentioning some preparatory results in Section 2, we will prove this the-
orem in Section 3. Note that similar to the one variable case, ζ(l1,...,lk)(q ,...,q )
k 1 k
also has the series expression
∞ ∞ (−logn )l1(−log(n +n ))l2···(−log(n +···+n ))lk
1 1 2 1 k
···
nq1(n +n )q2···(n +···+n )qk
nX1=1 nXk=1 1 1 2 1 k
for (l ,...,l ) ∈ (Z )k and (q ,...,q ) ∈ D .
1 k ≥0 1 k k
InSection 4, we will consider thecoefficients intheLaurent series ofζ (s)under
r
a certain restriction on the variables. Under this restriction, these coefficients can
be explicitly given by only using γ and ζ(l1,...,lk)(q ,...,q ). It implies that we
n k 1 k
can eliminate the role of γ from Theorem 1.3 under this restriction.
(n1,...,nk)
In Section 5, we will consider the Laurent series expansions at m ∈ Zr\(Z )r.
≥1
Since ζ (s) is singular on the hyperplanes
r
s = 1, s +s = 2,1,0,−2,−4,−6,...,
r r−1 r
and
s(j,r) = l (l ∈ Z, l ≤ r−j +1)
for 1 ≤ j ≤ r−2 ([1, Theorem 1]), the points m ∈ Zr \(Z )r are frequently on
≥1
these singular hyperplanes, and in many cases are the points of indeterminacy.
All of the aforementioned previous studies on non-positive integer points encoun-
tered this obstacle, and those studies discussed the limit values of ζ (s) when s
r
approaches m along various ways. In particular, the second-named author [11]
obtained a rather general result in which it allows a lot of flexibility how to ap-
proach the limit points, thoughhe did not arrive at theLaurent series expansions.
An example of this result will be mentioned just after Corollary 5.2.
We give Laurent series expansions at the points belonging to Zr\(Z )r, which
≥1
is the second main theorem in this paper. Define
(1.6) M (m) = M (m ,...,m ) := max{r −j −(m +···+m ) | 1 ≤ j ≤ r}.
r r 1 r j r
For l ≥ 2 and z ∈ C, let
l
Γ(s +z )
(1.7) F (s) := l l ζ (s ,...,s ,s +s +z ) for s = (s ,...,s ).
zl Γ(s ) l−1 1 l−2 l−1 l l 1 l
l
Then the result is the following
4 KOHJIMATSUMOTO, TOMOKAZU ONOZUKA,AND ISAOWAKABAYASHI
Theorem 1.4. Let m ∈ Zr\(Z )r. Then the coefficients of the Laurent series of
≥1
ζ (s) at s = m can be given explicitly in terms of the coefficients used in Theorem
r
1.3, ζ(n)(m)(m ≤ 0,n ≥ 0), and integrals
F(n1,...,nl)(k)Γ(−z )ζ(−z )dz
zl l l l
Z(Ml(k)+1−η)
where 2 ≤ l ≤ r, (n ,...,n ) ∈ (Z )l, k ∈ Zl \(Z )l, 0 < η < 1, and the path
1 l ≥0 ≥1
of integration is the vertical line ℜz = M (k)+1−η.
l l
A big difference from Theorem 1.3 is that, here, the coefficients may include
some integrals. This theorem will be proved in the first half of Section 5, and in
the second half of Section 5, we will discuss some cases when we may ignore the
contribution of integral terms.
The authors express their sincere gratitude to Professor Hidekazu Furusho,
Professor Yasushi Komori, Mr. Tomohiro Ikkai and Mr. Ryo Tanaka for valuable
comments and discussions.
2. Preliminaries
In the following sections, we use in the induction process the following key
formula proved by the first-named author (see [8, (12.7)] or [9, (4.4)]):
(2.1) ζ (s ,...,s )
r 1 r
1
= ζ (s ,...,s ,s +s −1)
r−1 1 r−2 r−1 r
s −1
r
M−1
−s
r
+ ζ (s ,...,s ,s +s +k )ζ(−k )
r−1 1 r−2 r−1 r r r
k
kXr=0(cid:18) r (cid:19)
1
+ I(s ,...,s ;M −η),
1 r
Γ(s )
r
where r ≥ 2, M is a positive integer, η is a small positive number, and
I(s ,...,s ;α)
1 r
1
= Γ(s +z )Γ(−z )ζ (s ,...,s ,s +s +z )ζ(−z )dz ,
r r r r−1 1 r−2 r−1 r r r r
2πi
Z(α)
whose path of integration is the vertical line ℜz = α. The formula (2.1), which
r
was proved by using the classical Mellin-Barnes integral formula, is valid in the
region where the above integral is convergent.
Forany points m ∈ Zr, there exists a sufficiently large M such that the integral
of(2.1)isanalytic atm(see Lemma 3.2). Hence this formulatells usthebehavior
of ζ (s ,...,s ) around the point s = m from the information on the behavior of
r 1 r
ζ .
r−1
Another key formula which we use is the so-called harmonic product formula.
This is used to show that the Laurent series expansion outside the domain of
convergence can be obtained from that inside the domain of convergence. The
harmonic product formula is obtained by just decomposing the summation. For
LAURENT SERIES EXPANSIONS OF MULTIPLE ZETA-FUNCTIONS 5
example, the product of the Euler-Zagier double zeta-function and the Riemann
zeta-function can be decomposed as follows:
(2.2)
ζ (s ,s )ζ(s )
2 1 2 3
1
=
ns1ns2ms3
n1<n20<m 1 2
X X
1
= + + + +
n1<n2<m n1<n2=m n1<m<n2 n1=m<n2 m<n1<n2! ns11ns22ms3
X X X X X
= ζ (s ,s ,s )+ζ (s ,s +s )+ζ (s ,s ,s )+ζ (s +s ,s )+ζ (s ,s ,s ).
3 1 2 3 2 1 2 3 3 1 3 2 2 1 3 2 3 3 1 2
Similarly we obtain
(2.3)
ζ(s )ζ (s ,s )
1 2 2 3
= ζ (s ,s ,s )+ζ (s +s ,s )+ζ (s ,s ,s )+ζ (s ,s +s )+ζ (s ,s ,s ).
3 1 2 3 2 1 2 3 3 2 1 3 2 2 1 3 3 2 3 1
The same method canbe applied to the decomposition of moregeneral product of
two Euler-Zagier multiple zeta-functions. For example, as a direct generalization
of (2.2), we have
(2.4) ζ (s ,...,s )ζ(s )
r−1 1 r−1 r
= ζ (s ,...,s )+ζ (s ,...,s ,s ,s )+···+ζ (s ,s ,...,s )
r 1 r r 1 r−2 r r−1 r r 1 r−1
+ζ (s ,...,s ,s +s )+ζ (s ,...,s ,s +s ,s )+···
r−1 1 r−2 r−1 r r−1 1 r−3 r−2 r r−1
+ζ (s +s ,s ,...,s ).
r−1 1 r 2 r−1
The most general form of the decomposition can be written as
(2.5) ζ (s ,...,s )ζ (s ,...,s )
j 1 j r−j j+1 r
= ζ (s ,...,s )+ζ (s ,...,s ,s ,s ,s ,s ,...,s )+···
r 1 r r 1 j−1 j+1 j j+2 j+3 r
+ζ (s ,s ,s ,...,s ,s ,s ,...,s )+···
r j+1 1 2 j−1 j j+2 r
+ζ (s ,...,s ,s ,...,s )
r j+1 r 1 j
+(the sum of ζ (l < r))
l
for 1 ≤ j ≤ r−1.
3. The Laurent series expansion at positive integer points
The main aim of this section is to prove Theorem 1.3. First, we determine the
order of the pole of the Euler-Zagier multiple zeta-function. For δ > 0, let
E (δ) = {(s ,...,s ) ∈ Cj | ℜs > 1−δ (1 ≤ l ≤ j)}.
j 1 j l
Lemma 3.1. For each j ∈ Z , there exists a function h (s ,...,s ), analytic
≥1 j 1 j+1
in the region E (δ ) with a sufficiently small positive constant δ (depending
j+1 j j
only on j), such that, for any r ∈ Z , the identity
≥2
(3.1)
ζ (s)
r
6 KOHJIMATSUMOTO, TOMOKAZU ONOZUKA,AND ISAOWAKABAYASHI
1
= ζ(s +···+s −(r −1))
1 r
(s −1)(s +s −2)···(s +···+s −(r −1))
r r−1 r 2 r
1
+ h (s ,s )
1 1 2
(s −1)(s +s −2)···(s +···+s −(r −2))
r r−1 r 3 r
1
+ h (s ,s ,s )
2 1 2 3
(s −1)(s +s −2)···(s +···+s −(r −3))
r r−1 r 4 r
+···
1
+ h (s ,...,s )
r−2 1 r−1
s −1
r
+h (s ,...,s ),
r−1 1 r
holds in the region E (δ ), especially at any point m ∈ (Z )r.
r r−1 ≥1
To prove Lemma 3.1, we use the part (ii) of the following lemma.
Lemma 3.2. (i) For any m ∈ Zr, if we choose M ≥ M (m)+1 (where M (m)
r r
is defined in (1.6)) and 0 < η < 1, the integral I(s ,...,s ;M −η) is analytic at
1 r
m.
(ii) In particular, I(s ,...,s ;1−η) is analytic at any point m ∈ (Z )r. In
1 r ≥1
fact, I(s ,...,s ;1−η) is analytic in the region E (1/r).
1 r r
(Proof of Lemma 3.2.) (i) Let F (M,η) be the set of (s ,...,s ) ∈ Cr satis-
r 1 r
fying
(3.2) ℜ(s +···+s ) > r −j −M +η (1 ≤ j ≤ r).
j r
By [8, Section 12], the integral I(s ,...,s ;M −η) is analytic on F (M,η). Con-
1 r r
sider the case s = m. We see that s = m ∈ F (M,η) if
r
(3.3) M > r −j −(m +···+m )+η (1 ≤ j ≤ r).
j r
It is obvious that under the conditions M ≥ M (m) + 1 and 0 < η < 1, the
r
inequality (3.3) holds. Hence m ∈ F (M,η) holds for M ≥ M (m)+1.
r r
(ii) If m ∈ (Z )r, then M (m) ≤ −1, so we can choose M = 1 in assertion
≥1 r
(i). This implies the first half of assertion (ii). When ℜs > 1 −δ (1 ≤ l ≤ r),
l
then (3.2) with M = 1 is satisfied if (r −j +1)(1−δ) > r −j −1+η, that is,
2 − η > (r − j + 1)δ for 1 ≤ j ≤ r. This is valid if we choose any δ satisfying
0 < δ < (2−η)/r, especially δ = 1/r. The second half of (ii) hence follows. (cid:3)
(Proof of Lemma 3.1.) We use the induction on r. First, we consider the case
r = 2. By (2.1) with M = 1, we have
1 1 1
ζ (s ,s ) = ζ(s +s −1)− ζ(s +s )+ I(s ,s ;1−η),
2 1 2 1 2 1 2 1 2
s −1 2 Γ(s )
2 2
because ζ(0) = −1/2. ¿From Lemma 3.2 (ii), the last term is analytic at s ∈
(Z )2. Furthermore the second term is also analytic at s ∈ (Z )2, since ζ(s)
≥1 ≥1
has only one pole at s = 1. Hence the case r = 2 is done. (The sum of the second
term and the third term gives h (s ,s ), with 0 < δ ≤ 1/2.)
1 1 2 1
Now we assume that (3.1) holds for r −1, that is, we have
(3.4)
ζ (s ,...,s ,s′ )
r−1 1 r−2 r−1
LAURENT SERIES EXPANSIONS OF MULTIPLE ZETA-FUNCTIONS 7
1
= ζ(s +···+s′ −(r −2))
(s′ −1)···(s +···+s +s′ −(r −2)) 1 r−1
r−1 2 r−2 r−1
1
+ h (s ,s )
(s′ −1)···(s +···+s +s′ −(r −3)) 1 1 2
r−1 3 r−2 r−1
+···
1
+ h (s ,...,s )
s′ −1 r−3 1 r−2
r−1
+h (s ,...,s′ ).
r−2 1 r−1
On the other hand, we use (2.1) with M = 1 to obtain
(3.5) ζ (s ,...,s )
r 1 r
1 1
= ζ (s ,...,s ,s +s −1)− ζ (s ,...,s ,s +s )
r−1 1 r−2 r−1 r r−1 1 r−2 r−1 r
s −1 2
r
1
+ I(s ,...,s ;1−η).
1 r
Γ(s )
r
Substituting (3.4) to (3.5) with s′ = s +s −1, we have
r−1 r−1 r
ζ (s ,...,s )
r 1 r
1 1
= ζ(s +···+s −(r −1))
1 r
s −1 (s +s −2)···(s +···+s −(r −1))
r (cid:26) r−1 r 2 r
1
+ h (s ,s )
1 1 2
(s +s −2)···(s +···+s −(r−2))
r−1 r 3 r
+···
1
+ h (s ,...,s )
r−3 1 r−2
(s +s −2)
r−1 r
+h (s ,...,s ,s +s −1)
r−2 1 r−2 r−1 r
(cid:27)
1 1
− ζ (s ,...,s ,s +s )+ I(s ,...,s ;1−η).
r−1 1 r−2 r−1 r 1 r
2 Γ(s )
r
Since (m ,...,m ,m + m ) ∈ D holds for m ∈ (Z )r, it is clear that
1 r−2 r−1 r r−1 ≥1
ζ (s ,...,s ,s + s ) is analytic at these points. Furthermore it follows
r−1 1 r−2 r−1 r
from Lemma 3.2 (ii) that the last term is also analytic. The term containing h
r−2
can be written as
h (s ,...,s ,s +s −1)−h (s ,...,s ,s ) h (s ,...,s )
r−2 1 r−2 r−1 r r−2 1 r−2 r−1 r−2 1 r−1
+ .
s −1 s −1
r r
The first term here is analytic at m ∈ (Z )r. (When s = 1, this term is to
≥1 r
be understood as the derivative of h with respect to the last variable.) Hence
r−2
putting
h (s ,...,s )
r−1 1 r
h (s ,...,s ,s +s −1)−h (s ,...,s )
r−2 1 r−2 r−1 r r−2 1 r−1
:=
s −1
r
1
− ζ (s ,...,s ,s +s )
r−1 1 r−2 r−1 r
2
8 KOHJIMATSUMOTO, TOMOKAZU ONOZUKA,AND ISAOWAKABAYASHI
1
+ I(s ,...,s ;1−η),
1 r
Γ(s )
r
which is analytic in E (δ ) for a sufficiently small δ > 0, we obtain Lemma
r r−1 r−1
(cid:3)
3.1.
(Proof of (1.5).) This is immediate from the formula (3.1) of Lemma 3.1, with
expanding the Riemann zeta factor to the Laurent series, and h factors to the
j
(cid:3)
Taylor series.
Next we prove Theorem 1.3.
(Proof of Theorem 1.3.) We use the induction on r. When r = 1, it follows
from (1.3) that Theorem 1.3 holds. We assume that Theorem 1.3 holds when the
number of variables is 1,2,...,r−1, and we prove that Theorem 1.3 holds for r.
Let m ∈ (Z )r. We define the case (C ) as
≥1 j
m > 1 if j = r,
(3.6) (C ) : r
j m > 1,m = m = ··· = m = 1 if 1 ≤ j ≤ r −1.
j j+1 j+2 r
(cid:26)
When m > 1, since m ∈ D holds, the series (1.1) is absolutely convergent at
r r
m. Therefore ζ (s) has the Taylor series expansion (1.4) at m, so we are done in
r
the case (C ).
r
Next we consider the case (C ), 1 ≤ j ≤ r − 1. We use the (down-going)
j
induction on j. Let 1 ≤ j ≤ r −1, assume that Theorem 1.3 holds in the cases
(C ),...,(C ), and we prove Theorem 1.3 for (C ). Our tool is the formula
j+1 r j
(2.5). The first term on the right-hand side of (2.5) is that we want to expand.
Coefficients in the Laurent series of the left-hand side and of the terms of ζ
l
(l < r) can be given explicitly by using γ and ζ(l1,...,lk)(q ,...,q ) by the
(n1,...,nk) k 1 k
assumption of induction on r. Coefficients in the Laurent series of the terms of ζ
r
except the first term on the right-hand side can be also given explicitly by using
γ and ζ(l1,...,lk)(q ,...,q ), by the assumption of induction on j, because
(n1,...,nk) k 1 k
in these terms, s is located at the l-th element of the r-tuple where l > j. Hence
j
in this case, Theorem 1.3 holds.
Therefore by induction, we obtain Theorem 1.3 for (C ) (j = 1,...,r − 1).
j
Finally, the only remaining case m = (1,...,1) is implied by (1.5). Thus we
complete the proof of Theorem 1.3. (We can also see the fact mentioned in com-
ment (ii) just after the statement of Theorem 1.3, by analyzing the above proof
(cid:3)
a little more carefully.)
Example 3.3. Here we explain the procedure given in the proof of Theorem 1.3,
by describing the case r = 3, m = (2,1,1). Assume s is close to 2, and s ,s are
1 2 3
close to 1. We first use (2.3):
(3.7) ζ (s ,s ,s ) = ζ(s )ζ (s ,s )−ζ (s +s ,s )
3 1 2 3 1 2 2 3 2 1 2 3
−ζ (s ,s ,s )−ζ (s ,s +s )−ζ (s ,s ,s ).
3 2 1 3 2 2 1 3 3 2 3 1
Since s is close to 2 and s +s is close to 3, the last two terms on the right-hand
1 1 3
side are in the domain of absolute convergence, so can be expanded by (1.4).
LAURENT SERIES EXPANSIONS OF MULTIPLE ZETA-FUNCTIONS 9
In particular, these are O(1). Next, we apply (2.2) to the third term on the
right-hand side:
(3.8) ζ (s ,s ,s ) = ζ(s ,s )ζ(s )−ζ (s ,s +s )
3 2 1 3 2 1 3 2 2 1 3
−ζ (s ,s ,s )−ζ (s +s ,s )−ζ (s ,s ,s ).
3 2 3 1 2 2 3 1 3 3 2 1
Since s is close to 2, all terms but the first one on the right-hand side are in the
1
domain of absolute convergence, hence can be expanded by (1.4) and O(1). The
first term can be expanded by (1.3) and (1.4), and can be written as
1
ζ(s ,s )ζ(s ) = ζ (s ,s ) +O(1) .
2 1 3 2 2 1
s −1
(cid:18) 3 (cid:19)
Therefore
ζ (s ,s )
2 2 1
(3.9) ζ (s ,s ,s ) = +O(1).
3 2 1 3
s −1
3
Tothesecondtermontheright-handsideof (3.7),weapplythesimplestharmonic
product formula
(3.10) ζ(s )ζ(s ) = ζ (s ,s )+ζ (s ,s )+ζ(s +s )
1 2 2 1 2 2 2 1 1 2
to obtain
ζ (s +s ,s ) = ζ(s +s )ζ(s )−ζ (s ,s +s )−ζ(s +s +s ).
2 1 2 3 1 2 3 2 3 1 2 1 2 3
The second and the third terms on the right-hand side are in the domain of
absolute convergence, and so
ζ(s +s )
1 2
(3.11) ζ (s +s ,s ) = +O(1).
2 1 2 3
s −1
3
Finally, since (s ,s ) is close to (1,1), we use (1.5) to obtain
2 3
1 1
(3.12) ζ(s )ζ (s ,s ) = ζ(s ) +A(s ,s ) ,
1 2 2 3 1 2 3
s −1 s +s −2
3 (cid:18) 2 3 (cid:19)
where
∞ ∞
A(s ,s ) = γ (s −1)n1(s −1)n2.
2 3 (n1,n2) 2 3
n1=0n2=0
X X
By the above argument it is clear that ζ (s ,s ,s ) can be expanded to the Lau-
3 1 2 3
rent series around the point (2,1,1), and especially
ζ(s )
1
(3.13) ζ (s ,s ,s ) =
3 1 2 3
(s −1)(s +s −2)
3 2 3
ζ(s )A(s ,s )−ζ(s +s )−ζ (s ,s )
1 2 3 1 2 2 2 1
+ +O(1),
s −1
3
where the terms on the numerators are all holomorphic there. Expanding the
numerators and the O(1) term to the Taylor series, we obtain an example of
comment (ii) after the statement of Theorem 1.3.
10 KOHJIMATSUMOTO, TOMOKAZU ONOZUKA,AND ISAOWAKABAYASHI
4. The Laurent series expansion at positive integer points under a
certain additional restriction
Inthissection, weconsider theLaurentseriesexpansionof (1.1)underacertain
additional restriction on the variables.
Definition 4.1. Let m = (m ,...,m ) ∈ (Z )r. Let h be the number of m
1 r ≥1 j
which is equal to 1, and denote those m s by m ,...,m . By the restricted
j a1 ah
Laurent series expansion of ζ (s) at m we mean the Laurent series expansion of
r
ζ (s) at m with the restriction s = ··· = s = s.
r a1 ah
If we add the above restriction, then the coefficients of the Laurent series can
be given in the following simpler form than Theorem 1.3.
Theorem 4.2. Let r,m ,...,m be positive integers. The coefficients in the
1 r
restricted Laurent series of ζ (s) at s = m can be given explicitly by using γ and
r n
ζ(l1,...,lk)(q ,...,q ) for n ≥ 0, 1 ≤ k ≤ r, (l ,...,l ) ∈ (Z )k and (q ,...,q ) ∈
k 1 k 1 k ≥0 1 k
D ∩(Z )k.
k ≥1
(Proof of Theorem 4.2.) The proof of Theorem 4.2 is quite similar to the
proof of Theorem 1.3, so we omit the details. The main difference is the last step
of induction, i.e. the case m = (1,...,1). In this step, we used (1.5) in the proof
of Theorem 1.3. However, in the proof of Theorem 4.2, we can not use (1.5), since
its coefficients include γ . Therefore, we have to use a different method.
(n1,...,nk)
The idea is to use (2.4) once more. In this case, since s = ··· = s = s, we
1 r
can simplify (2.4) as follows:
ζ (s,...,s)ζ(s) = rζ (s,...,s)
r−1 r
+ζ (s,...,s,2s)+ζ (s,...,s,2s,s)+···+ζ (2s,s,...,s).
r−1 r−1 r−1
The first term of the right-hand side is that we want to expand, and other terms
(cid:3)
can be expanded by assumption. Hence we obtain Theorem 4.2.
¿From (1.5) and Theorem 4.2, we can easily deduce the following corollary.
Corollary 4.3. For any non-negative integer N, the sum
γ
(n1,...,nr)
n1+·X··+nr=N
n1,...,nr≥0
can be written explicitly by using γ and ζ(l1,...,lk)(q ,...,q ) for n ≥ 0, 1 ≤ k ≤ r,
n k 1 k
(l ,...,l ) ∈ (Z )k and (q ,...,q ) ∈ D ∩(Z )k.
1 k ≥0 1 k k ≥1
(Proof of Corollary 4.3.) By (1.5), we have
ζ (s,...,s)
r
∞ ∞
1 1 1 1
= + ··· γ (s−1)n1+···+nr
r!(s−1)r (r −1)!(s−1)r−1 (n1,...,nr)
nX1=0 nXr=0
∞
1 1 1 1
= + γ (s−1)N.
r!(s−1)r (r −1)!(s−1)r−1 (n1,...,nr)
NX=0n1+·X··+nr=N
n1,...,nr≥0