TAMKANG JOURNAL OF MATHEMATICS
Volume 44, Number 2, 123-129, Summer 2013
doi:10.5556/j.tkjm.44.2013.1128
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Available online at http://journals.math.tku.edu.tw/
SOME NEW PROOFS OF THE SUM FORMULA AND
RESTRICTED SUM FORMULA
C. Y. CHANG
Abstract. The sum formula is a basic identify of multiple zeta values that expresses a
Riemann-zeta value as a homogeneous sum of multiple zeta values of given depth and
weight. This formula was already known to Euler in the depth two case. Conjectured in
the early 1990s, for higher depth and then proved by Granville and Zagier independently.
Restricted sum formula was given in Eie [2]. In this paper, we present some new proofs
of those formulas.
1. Introduction
Multiple zeta values are special values of the multi-variable analytic function
ζ(s 1 , s 2 , . . . , s n ) =
X
1≤n1 <n2 <···<nk
1
s
s
s
n 11 n 22 · · · n kk
at positive integers s i ≥ 1, s k ≥ 2, 1 ≤ i ≤ k, the number k and the number |s| = s 1 + s 2 + · · · + s k
is called the depth and the weight of ζ(s 1 , s 2 , . . . , s n ), respectively. The most well-known sum
formula
X
ζ(s 1 , s 2 , . . . , s k + 1) = ζ(n + 1)
si ≥1, ∀i
|s|=n
suggesting intriguing connection between Riemann Zeta values and the multiple zeta values.
This formula was first obtained by Euler where k = 2, known as Euler’s sum formula
n−1
X
ζ(n − i , i ) = ζ(n).
i =2
It’s general form was conjectured by M. Schmidt [7] and C. Moen [6] around 1994. A Granville
[4] proved the general case in 1997.
For the convenience, we denote {1}k as k repetitions of 1. The sum formula gives ζ({1}n−1 , 2) =
ζ(n+1). In general we have the Drinfeld duality ζ({1}m−1 , n+1) = ζ({1}n−1 , m+1). In this paper,
2010 Mathematics Subject Classification. 11M.
Key words and phrases. Drinfeld integral, multiple zeta values, sum formula.
123
124
C. Y. CHANG
we also consider the more general form
X
ζ({1}n , s 1 , s 2 , . . . , s k + 1)
si ≥2, ∀i
|s|=m
known as the restricted sum formula and give another proof of the restricted sum formula
X
ζ({1}n , s 1 , s 2 , . . . , s k + 1) =
X
ζ(t 1 , t 2 , . . . , t p+1 + (m − k) + 1)
si ≥1, ∀i
|t|=n+k
si ≥1, ∀i
|s|=m
where m, k are position integers with, m ≥ k and n is non-negative integer. The restricted
sum formula was originally due to M. Eie [2].
2. The double integral representation of a sum
Drinfeld integrals give a multiple zeta values an integral representation. For example
m
ζ({1} , n + 1) =
m+1
Y
Z
0<t 1 <t 2 <···<t m+n+1 <1 j =1
dtj
1−tj
m+n+1
Y dt
k=m+2
k
tk
.
In general we have as in [3]
ζ(α0 , α1 , . . . , αk + 1) =
where

d tk



1 − tk
Ωk =
d tk



tk
Z
0<t 1 <t 2 <···<t |α|+1
Ω1 Ω2 · · · Ω|α|+1
k = 1, α0 + 1, α0 + α1 + 1, . . . , α0 + α1 + · · · + αk−1 + 1,
otherwise.
For example we have
ζ(2, 2, 3) =
d t1 d t2 d t3 d t4 d t5 d t6 d t7
.
1
0<t 1 <t 2 <···<t 7 <1 − t 1 t 2 1 − t 3 t 4 1 − t 5 t 6 t 7
Z
Now, fix t 1 , t α0 +1 , . . . , t α0 +α1 +···+αk−1 +1 and t |α|+1 and integrate with respect to the remainning
variables, we obtain that
ζ(α0 , α1 , . . . , αk + 1) =
Z
k+1
Y
0<t 1 <t 2 <···<t k+2 <1 j =1
dtj
1
i − t j (α j −1 − 1)!
log
³ t j +1 ´α j −1 −1 d t
tj
k+2
t k+2
.
Denote α j − 1 = β j , 0 ≤ j ≤ k, and consider all the nonnegative integers β0 , β1 , . . . , βk with
|β| = m, we have
t k+2 ´m
t2
t3
t k+2 ´m
1 ³
1 ³
log
log + log + · · · + log
=
m!
t1
m!
t1
t2
t k+1
SOME NEW PROOFS OF THE SUM FORMULA AND RESTRICTED SUM FORMULA
=
125
³
1
t 2 ´β0 ³
t 3 ´β1 ³
t k+2 ´βk
log
.
log
· · · log
t1
t2
t k+1
|β|=m β0 !β1 ! · · · βk !
X
Hence we get
1
ζ(α0 , α1 , . . . , αk + 1) =
m!
|α|=m+k+1
X
Z
0<t 1 <t 2 <···<t k+2 <1
³
log
Y d t j d t k+2
t k+2 ´m k+1
.
t1
j =1 i − t j t k+2
Now fixed t 1 , t k+2, write it as t 1, t 2 , and integrate with respect to the remainning variables we
obtain the following theorem.
Theorem 1. Let m, k be non-negative integers. Then
Z
³
X
1
1 − t 1 ´k ³
t 2 ´m d t 1 d t 2
ζ(α0 , α1 , . . . , αk + 1) =
log
.
log
m!k! 0<t1 <t2 <1
1 − t2
t1 1 − t1 t2
|α|=m+k+1
Based on the above theorem, we are able to prove the sum formula via changes of variables in
the integral.
First proof of the sum formula :
Let
x = log
we have
d xd y =
and
Z
0<t 1 <t 2 <1
³
d t1 d t2
,
1 − t1 t2
1
1 − t1
log
y = log
1 − t1
1
= log x
,
1 − t2
e + e y − e x+y
t 2 ´m d t 1 d t 2
1 − t 1 ´k ³
=
log
log
1 − t2
t1 1 − t1 t2
x
y
where D = {(x, y) | x > 0, y > 0, e + e > e
x+y
t2
t1
Z
D
}.
³
y m log
´k
1
d xd y
e x + e y − e x+y
On the other hand, also consider the transformation
x → y,
y → x.
Then by the symmetry of D, we get
Z
Z
³
´k
´k
³
1
1
m
m
y log x
d
xd
y
=
d xd y
x
log
e + e y − e x+y
e x + e y − e x+y
D
ZD
³
1 − t 1 ´k d t 1 d t 2
1 ´m ³
.
log
=
log
1 − t1
1 − t2 1 − t1 t2
0<t 1 <t 2 <1
But we already know that
ζ({1}m+k , 2) =
Z
0<t 1 <t 2 <···<t k+2 <1
m+k+1
Y
j =1
d t j d t m+k+2
1 − t j t m+k+2
126
C. Y. CHANG
1
=
m!k!
Z
0<t 1 <t 2 <1
³
log
1 ´m ³
1 − t 1 ´k d t 1 d t 2
.
log
1 − t1
1 − t2 1 − t1 t2
So we get
ζ(α0 , α1 , . . . , αk + 1) = ζ({1}m+k , 2) = ζ(m + k + 2)
X
|α|=m+k+1
by Drinfeld duality theorem.
3. Euler sums with two branches
Euler sums with two branches was introduced in Eie [2]. For a pair of integers p, q ≥ 0,
and positive integers n ≥ 2. We let
E n (p, q) =
∞ 1 X
X
X
1
1
,
n
k 0 =1 k 0 k k 1 k 2 · · · k q l l 1 l 2 · · · l q
where (k 0 , k 1 , . . . , k q ) ∈ N q+1 , with k 0 ≥ k 1 ≥ · · · ≥ k q ≥ 1 and (l 1 , . . . , l p ) ∈ N p , with k 0 > l 1 > l 2 >
· · · > l q ≥ 1.
On the other hand we also have
E n (p, q) =
X
1
X
n−1
k∈N p+1 u∈N q σ1 σ2 · · · σp σp+1 τ1 τ2 · · · τq (τq
+ σp+1 )
where σ j = k 1 + k 2 + · · · + k j , τl = u 1 + · · · + u l .
So that we have the following Drinfeld integral representation:
E n (p, q) =
q
d u λ d t p+n
d tk Y
,
j =1 1 − t j k=p+2 t k λ=1 1 − u λ t p+n
Z p+1
Y dtj
D
n+p−1
Y
where D is a region of R p+q+n defined by
{0 < t 1 < t 2 < · · · < t p+n < 1,
0 < u 1 < u 2 < · · · < u q < t p+n }.
Now fix t p+1 , p+n , as t 1 , t 2 , and integrate with respect to the remainning variables, we get
E n (p, q) =
1
p!q!(n − 2)!
Z
0<t 1 <t 2 <1
³
log
1 ´p ³
1 ´q ³
t 2 ´n−2 d t 1 d t 2
.
log
log
1 − t1
1 − t2
t1
1 − t1 t2
Now from the double integral representation for a sum and E n (p, q), we get
X
|α|=m+k+1
ζ(α0 , α1 , α2 , . . . , αk+1 ) =
X
(−1)p E m+2 (p, q).
p+q=k
We need the following decomposition theorem for E m+2 (p, q).
SOME NEW PROOFS OF THE SUM FORMULA AND RESTRICTED SUM FORMULA
Theorem 2. For nonnegative integers m, p, q, we have
Ã
!
p+q+1
X r −1
X
ζ(α1 , . . . , αr + m + 1).
E m+2 (p, q) =
p |α|=p+q+1
r =p+1
We begin with the integral representation.
Proof.
1
E m+2 (p, q) =
p!q!m!
Z
0<t 1 <t 2 <1
³
log
1 ´p ³
1 ´q ³
t 2 ´m d t 1 d t 2
.
log
log
1 − t1
1 − t2
t1 1 − t1 t2
Express
³
à !
q
1 ´q−b ³
1 − t 1 ´b
1 ´q X
q ³
log
=
log
log
,
1 − t2
1 − t1
1 − t2
b=0 b
the resulting integral represents the sum of multiple zeta value
Ã
!
q
X
X
p +q −b
ζ({1}p+q−b , α0 , α1 , . . . , αb + 1).
p
b=0
|α|=b+m+1
Now let u 1 = 1 − t 2 , u 2 = 1 − t 1 , the integral is transformed into
q
X
Z
³
1 − u 1 ´m ³
u 2 ´b ³
1 ´p+q−b d u 1 d u 2
1
log
log
log
1 − u2
u1
u2
1 − u1 u2
b=0 p!q!(q − b)!m! 0<u1 <u2 <1
Ã
!
q
X
X
p +q −b
=
ζ(α0 , α1 , . . . , αm + p + q − b + 1).
p
b=0
|α|=b+m+1
In the final, we apply Ohno’s generalization of duality theorem and sum formula with
k = ({1}m , p + q − b + 2),
k ′ = ({1}p+q−b , m + 2).
We reduce the above sum formula to
Ã
!
p+q+1
X r −1
X
ζ(α1 , α2 , . . . , αr + m + 1).
p |α|=p+q+1
r =p+1
New let
St =
X
ζ(α1 , . . . , αt + m + 1),
t = 1, 2, . . . , r + 1.
|α|=r +1
we get the following from the decomposition theorem
E m+2 (0, r ) = S r +1 + S r + · · · + S 3 + s 2 + s 1
E m+2 (1, r − 1) = r S r +1 + (r − 1)S r + · · · + 2S 3 + s 2
à !
Ã
!
r
r −1
E m+2 (2, r − 2) =
S r +1 +
Sr + · · · + S3
2
2
127
128
C. Y. CHANG
..
.
E m+2 (r, 0) = S r +1 .
So we get
formula.
P
p+q=r
(−1)p E m+2 (p, q) = S 1 = ζ(m + r + 2). This gives a second proof of the sum
4. The restricted sum formula
The restricted sum can be expressed as an double integral:
X
ζ({1}p , α0 , α1 , . . . , αq + 1)
|α|=q+r +1
1
=
p!q!r !
Z
0<t 1 <t 2 <1
³
log
1 ´p ³
1 − t 1 ´q ³
t 2 ´r d t 1 d t 2
.
log
log
1 − t1
1 − t2
t1 1 − t1 t2
Make the change of variables:
x = log
1
1 − t1
y = log
t2
t1
the above integral is transformed into
1
p!q!r !
Z
xp yr
D2
³
´q
1
d xd y.
e x + e y − e x+y
1−u
Change coordinate again x = log 1−u12 , y = log u12 .
We get the following integral representation for the restricted sum:
1
p!q!r !
Z
0<u1 <u2 <1
³
log
1 − u 1 ´p ³
u 2 ´q ³
1 ´r d u 1 d u 2
log
log
1 − u2
u1
u2 1 − u1 u2
which is
X
ζ(c 0 , c 1 , . . . , c p + r + 1).
|c|=p+q+1
Therefore, the restricted sum formula is proved. In the following, we given another proof of
the same formula. From the double integral formula for both
X
ζ({1}p , α0 , α1 , . . . , αq + 1)
|α|=q+r +1
and E m (p, q) we get
Ã
!
q
X
p +b
(−1)b E r +2 (p + b, q − b).
ζ({1} , α0 , α1 , . . . , αq + 1) =
b
|α|=q+r +1
b=0
X
p
SOME NEW PROOFS OF THE SUM FORMULA AND RESTRICTED SUM FORMULA
129
Substitute the decomposition of E r +2 (p + b, q − b) as
Ã
!
p+q+1
X
X
g −1
S g , with S g =
ζ(α1 , . . . , αg + r + 1)
|α|=p+q+1
g =p+b+1 p + b
into the identity
Ã
!
q
X
p +b
ζ({1} , α0 , α1 , . . . , αq + 1) =
(−1)b E r +2 (p + b, q − b).
b
|α|=q+r +1
b=0
X
p
We get
Ã
!
Ã
!
q
p+q+1
X
X
p
+
b
g
−
1
ζ({1}p , α0 , . . . , αp+1 ) =
(−1)b
Sg .
p
|α|=q+r +1
b=0
g =p+b+1 p + b
X
Exchange the order of summation, we get
Ã
!
Ã
!
g −p−1
p+q+1
X g −p −1
X g −1
(−1)b .
Sg
b
p
g =p+1
b=0
Again, the inner sum is zero unless g − p − 1 = 0. Therefore the total sum is S p+1 or
P
ζ(c 0 , c 1 , . . . , c p + r + 1).
|c|=p+q+1
References
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129(2009), 908–921.
[3] M. Eie and C. S. Wei, A short proof for the sum formula and it’s generalization, Arch. Math., 91(2008), 330–338.
[4] A. Granville, A decomposition of Riemann’s zeta-function, in: Analytic Number Theory, Kyoto, 1996, in: London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, (1997), 95–101.
[5] M. E. Hoffman, Multiple harmonic series, Pacific J. Math., 152(1992), 275–290.
[6] M. E. Hoffman and C. Moen, Sums of triple harmonic series, J. Number Theory, 60(1996), 329–331.
[7] C. Markett, Triple sums and the Riemann zeta function, J. Number Theory, 48(1994), 113–132.
[8] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory,
74(1999), 39–43.
[9] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II
(Paris, 1992), Progr. Math., 120, Basel, Boston, Berlin: Birkhauser, 497–512, 1994.
Department of Applied Mathematics, Tatung University. No.40, Sec. 3, Zhongshan N. Rd., Taipei City 104, Taiwan
(R.O.C.).
E-mail: [email protected]