In mathematics , the multiple zeta functions are generalisations of the Riemann zeta function , defined by
ζ
(
s
1
,
…
,
s
k
)
=
∑
n
1
>
n
2
>
⋯
>
n
k
>
0
1
n
1
s
1
⋯
n
k
s
k
=
∑
n
1
>
n
2
>
⋯
>
n
k
>
0
∏
i
=
1
k
1
n
i
s
i
,
{\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^{s_{1}}\cdots n_{k}^{s_{k}}}}=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ \prod _{i=1}^{k}{\frac {1}{n_{i}^{s_{i}}}},\!}
and converge when Re(s 1 ) + ... + Re(s i ) > i for all i . Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When s 1 , ..., s k are all positive integers (with s 1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums .
The k in the above definition is named the "length" of a MZV, and the n = s 1 + ... + s k is known as the "weight".[ 1]
The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the number of repetitions. For example,
ζ
(
2
,
1
,
2
,
1
,
3
)
=
ζ
(
{
2
,
1
}
2
,
3
)
{\displaystyle \zeta (2,1,2,1,3)=\zeta (\{2,1\}^{2},3)}
Two parameters case
In the particular case of only two parameters we have (with s>1 and n,m integer):[ 2]
ζ
(
s
,
t
)
=
∑
n
>
m
≥
1
1
n
s
m
t
=
∑
n
=
1
∞
1
n
s
∑
m
=
1
n
−
1
1
m
t
=
∑
n
=
1
∞
1
(
n
+
1
)
s
∑
m
=
1
n
1
m
t
{\displaystyle \zeta (s,t)=\sum _{n>m\geq 1}\ {\frac {1}{n^{s}m^{t}}}=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\sum _{m=1}^{n-1}{\frac {1}{m^{t}}}=\sum _{n=1}^{\infty }{\frac {1}{(n+1)^{s}}}\sum _{m=1}^{n}{\frac {1}{m^{t}}}}
ζ
(
s
,
t
)
=
∑
n
=
1
∞
H
n
,
t
(
n
+
1
)
s
{\displaystyle \zeta (s,t)=\sum _{n=1}^{\infty }{\frac {H_{n,t}}{(n+1)^{s}}}}
where
H
n
,
t
{\displaystyle H_{n,t}}
are the generalized harmonic numbers .
Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler :
∑
n
=
1
∞
H
n
(
n
+
1
)
2
=
ζ
(
2
,
1
)
=
ζ
(
3
)
=
∑
n
=
1
∞
1
n
3
,
{\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{(n+1)^{2}}}=\zeta (2,1)=\zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}},\!}
where H n are the harmonic numbers .
Special values of double zeta functions, with s > 0 and even, t > 1 and odd, but s+t=2N+1 (taking if necessary ζ (0) = 0):[ 2]
ζ
(
s
,
t
)
=
ζ
(
s
)
ζ
(
t
)
+
1
2
[
(
s
+
t
s
)
−
1
]
ζ
(
s
+
t
)
−
∑
r
=
1
N
−
1
[
(
2
r
s
−
1
)
+
(
2
r
t
−
1
)
]
ζ
(
2
r
+
1
)
ζ
(
s
+
t
−
1
−
2
r
)
{\displaystyle \zeta (s,t)=\zeta (s)\zeta (t)+{\tfrac {1}{2}}{\Big [}{\tbinom {s+t}{s}}-1{\Big ]}\zeta (s+t)-\sum _{r=1}^{N-1}{\Big [}{\tbinom {2r}{s-1}}+{\tbinom {2r}{t-1}}{\Big ]}\zeta (2r+1)\zeta (s+t-1-2r)}
class="wikitable "
Note that if
s
+
t
=
2
p
+
2
{\displaystyle s+t=2p+2}
we have
p
/
3
{\displaystyle p/3}
irreducibles, i.e. these MZVs cannot be written as function of
ζ
(
a
)
{\displaystyle \zeta (a)}
only.[ 3]
Three parameters case
In the particular case of only three parameters we have (with a>1 and n,j,i integer):
ζ
(
a
,
b
,
c
)
=
∑
n
>
j
>
i
≥
1
1
n
a
j
b
i
c
=
∑
n
=
1
∞
1
(
n
+
2
)
a
∑
j
=
1
n
1
(
j
+
1
)
b
∑
i
=
1
j
1
(
i
)
c
=
∑
n
=
1
∞
1
(
n
+
2
)
a
∑
j
=
1
n
H
i
,
c
(
j
+
1
)
b
{\displaystyle \zeta (a,b,c)=\sum _{n>j>i\geq 1}\ {\frac {1}{n^{a}j^{b}i^{c}}}=\sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{j=1}^{n}{\frac {1}{(j+1)^{b}}}\sum _{i=1}^{j}{\frac {1}{(i)^{c}}}=\sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{j=1}^{n}{\frac {H_{i,c}}{(j+1)^{b}}}}
The above MZVs satisfy the Euler reflection formula:
ζ
(
a
,
b
)
+
ζ
(
b
,
a
)
=
ζ
(
a
)
ζ
(
b
)
−
ζ
(
a
+
b
)
{\displaystyle \zeta (a,b)+\zeta (b,a)=\zeta (a)\zeta (b)-\zeta (a+b)}
for
a
,
b
>
1
{\displaystyle a,b>1}
Using the shuffle relations, it is easy to prove that:[ 3]
ζ
(
a
,
b
,
c
)
+
ζ
(
a
,
c
,
b
)
+
ζ
(
b
,
a
,
c
)
+
ζ
(
b
,
c
,
a
)
+
ζ
(
c
,
a
,
b
)
+
ζ
(
c
,
b
,
a
)
=
ζ
(
a
)
ζ
(
b
)
ζ
(
c
)
+
2
ζ
(
a
+
b
+
c
)
−
ζ
(
a
)
ζ
(
b
+
c
)
−
ζ
(
b
)
ζ
(
a
+
c
)
−
ζ
(
c
)
ζ
(
a
+
b
)
{\displaystyle \zeta (a,b,c)+\zeta (a,c,b)+\zeta (b,a,c)+\zeta (b,c,a)+\zeta (c,a,b)+\zeta (c,b,a)=\zeta (a)\zeta (b)\zeta (c)+2\zeta (a+b+c)-\zeta (a)\zeta (b+c)-\zeta (b)\zeta (a+c)-\zeta (c)\zeta (a+b)}
for
a
,
b
,
c
>
1
{\displaystyle a,b,c>1}
This function can be seen as a generalization of the reflection formulas.
Symmetric sums in terms of the zeta function
Let
S
(
i
1
,
i
2
,
⋯
,
i
k
)
=
∑
n
1
≥
n
2
≥
⋯
n
k
≥
1
1
n
1
i
1
n
2
i
2
⋯
n
k
i
k
{\displaystyle S(i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}\geq n_{2}\geq \cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}
, and for a partition
Π
=
{
P
1
,
P
2
,
…
,
P
l
}
{\displaystyle \Pi =\{P_{1},P_{2},\dots ,P_{l}\}}
of the set
{
1
,
2
,
…
,
k
}
{\displaystyle \{1,2,\dots ,k\}}
, let
c
(
Π
)
=
(
|
P
1
|
−
1
)
!
(
|
P
2
|
−
1
)
!
⋯
(
|
P
l
|
−
1
)
!
{\displaystyle c(\Pi )=(\left|P_{1}\right|-1)!(\left|P_{2}\right|-1)!\cdots (\left|P_{l}\right|-1)!}
. Also, given such a
Π
{\displaystyle \Pi }
and a k-tuple
i
=
{
i
1
,
.
.
.
,
i
k
}
{\displaystyle i=\{i_{1},...,i_{k}\}}
of exponents, define
∏
s
=
1
l
ζ
(
∑
j
∈
P
s
i
j
)
{\displaystyle \prod _{s=1}^{l}\zeta (\sum _{j\in P_{s}}i_{j})}
.
The relations between the
ζ
{\displaystyle \zeta }
and
S
{\displaystyle S}
are:
S
(
i
1
,
i
2
)
=
ζ
(
i
1
,
i
2
)
+
ζ
(
i
1
+
i
2
)
{\displaystyle S(i_{1},i_{2})=\zeta (i_{1},i_{2})+\zeta (i_{1}+i_{2})}
and
S
(
i
1
,
i
2
,
i
3
)
=
ζ
(
i
1
,
i
2
,
i
3
)
+
ζ
(
i
1
+
i
2
,
i
3
)
+
ζ
(
i
1
,
i
2
+
i
3
)
+
ζ
(
i
1
+
i
2
+
i
3
)
{\displaystyle S(i_{1},i_{2},i_{3})=\zeta (i_{1},i_{2},i_{3})+\zeta (i_{1}+i_{2},i_{3})+\zeta (i_{1},i_{2}+i_{3})+\zeta (i_{1}+i_{2}+i_{3})}
Theorem 1(Hoffman)
For any real
i
1
,
⋯
,
i
k
>
1
,
{\displaystyle i_{1},\cdots ,i_{k}>1,}
,
∑
σ
∈
∑
k
S
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \sum _{k}}S(i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}c(\Pi )\zeta (i,\Pi )}
.
Proof. Assume the
i
j
{\displaystyle i_{j}}
are all distinct. (There is not loss of generality, since we can take limits.) The left-hand side can be written as
∑
σ
∑
n
1
≥
n
2
≥
⋯
≥
n
k
≥
1
1
n
i
1
σ
(
1
)
n
i
2
σ
(
2
)
⋯
n
i
k
σ
(
k
)
{\displaystyle \sum _{\sigma }\sum _{n_{1}\geq n_{2}\geq \cdots \geq n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}}
. Now thinking on the symmetric
group
∑
k
{\displaystyle \sum _{k}}
as acting on k-tuple
n
=
(
1
,
⋯
,
k
)
{\displaystyle n=(1,\cdots ,k)}
of positive integers. A given k-tuple
n
=
(
n
1
,
⋯
,
n
k
)
{\displaystyle n=(n_{1},\cdots ,n_{k})}
has an isotropy group
∑
k
(
n
)
{\displaystyle \sum _{k}(n)}
and an associated partition
Λ
{\displaystyle \Lambda }
of
(
1
,
2
,
⋯
,
k
)
{\displaystyle (1,2,\cdots ,k)}
:
Λ
{\displaystyle \Lambda }
is the set of equivalence classes of the relation
given by
i
∼
j
{\displaystyle i\sim j}
iff
n
i
=
n
j
{\displaystyle n_{i}=n_{j}}
, and
∑
k
(
n
)
=
{
σ
∈
∑
k
:
σ
(
i
)
∼
∀
i
}
{\displaystyle \sum _{k}(n)=\{\sigma \in \sum _{k}:\sigma (i)\sim \forall i\}}
. Now the term
1
n
i
1
σ
(
1
)
n
i
2
σ
(
2
)
⋯
n
i
k
σ
(
k
)
{\displaystyle {\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}}
occurs on the left-hand side of
∑
σ
∈
∑
k
S
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \sum _{k}}S(i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}c(\Pi )\zeta (i,\Pi )}
exactly
|
∑
k
(
n
)
|
{\displaystyle \left|\sum _{k}(n)\right|}
times. It occurs on the right-hand side in those terms corresponding to partitions
Π
{\displaystyle \Pi }
that are refinements of
Λ
{\displaystyle \Lambda }
: letting
⪰
{\displaystyle \succeq }
denote refinement,
1
n
i
1
σ
(
1
)
n
i
2
σ
(
2
)
⋯
n
i
k
σ
(
k
)
{\displaystyle {\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}}
occurs
∑
Π
⪰
Λ
(
Π
)
{\displaystyle \sum _{\Pi \succeq \Lambda }(\Pi )}
times. Thus, the conclusion will follow if
|
∑
k
(
n
)
|
=
∑
Π
⪰
Λ
c
(
Π
)
{\displaystyle \left|\sum _{k}(n)\right|=\sum _{\Pi \succeq \Lambda }c(\Pi )}
for any k-tuple
n
=
{
n
1
,
⋯
,
n
k
}
{\displaystyle n=\{n_{1},\cdots ,n_{k}\}}
and associated partition
Λ
{\displaystyle \Lambda }
.
To see this, note that
c
(
Π
)
{\displaystyle c(\Pi )}
counts the permutations having cycle-type specified by
Π
{\displaystyle \Pi }
: since any elements of
∑
k
(
n
)
{\displaystyle \sum _{k}(n)}
has a unique cycle-type specified by a partition that refines
Λ
{\displaystyle \Lambda }
, the result follows.[ 4]
For
k
=
3
{\displaystyle k=3}
, the theorem says
∑
σ
∈
∑
3
S
(
i
σ
(
1
)
,
i
σ
(
2
)
,
i
σ
(
3
)
)
=
ζ
(
i
1
)
ζ
(
i
2
)
ζ
(
i
3
)
+
ζ
(
i
1
+
i
2
)
ζ
(
i
3
)
+
ζ
(
i
1
)
ζ
(
i
2
+
i
3
)
+
ζ
(
i
1
+
i
3
)
ζ
(
i
2
)
+
2
ζ
(
i
1
+
i
2
+
i
3
)
{\displaystyle \sum _{\sigma \in \sum _{3}}S(i_{\sigma (1)},i_{\sigma (2)},i_{\sigma (3)})=\zeta (i_{1})\zeta (i_{2})\zeta (i_{3})+\zeta (i_{1}+i_{2})\zeta (i_{3})+\zeta (i_{1})\zeta (i_{2}+i_{3})+\zeta (i_{1}+i_{3})\zeta (i_{2})+2\zeta (i_{1}+i_{2}+i_{3})}
for
i
1
,
i
2
,
i
3
>
1
{\displaystyle i_{1},i_{2},i_{3}>1}
. This is the main result of.[ 5]
Having
ζ
(
i
1
,
i
2
,
⋯
,
i
k
)
=
∑
n
1
>
n
2
>
⋯
n
k
≥
1
1
n
1
i
1
n
2
i
2
⋯
n
k
i
k
{\displaystyle \zeta (i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}>n_{2}>\cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}
. To state the analog of Theorem 1 for the
ζ
′
s
{\displaystyle \zeta 's}
, we require one bit of notation. For a partition
Π
=
{
P
1
,
⋯
,
P
l
}
{\displaystyle \Pi =\{P_{1},\cdots ,P_{l}\}}
or
{
1
,
2
⋯
,
k
}
{\displaystyle \{1,2\cdots ,k\}}
, let
c
~
(
Π
)
=
(
−
1
)
k
−
l
c
(
Π
)
{\displaystyle {\tilde {c}}(\Pi )=(-1)^{k-l}c(\Pi )}
.
Theorem 2(Hoffman)
For any real
i
1
,
⋯
,
i
k
>
1
{\displaystyle i_{1},\cdots ,i_{k}>1}
,
∑
σ
∈
∑
k
ζ
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
~
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \sum _{k}}\zeta (i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}{\tilde {c}}(\Pi )\zeta (i,\Pi )}
.
Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now
∑
σ
∑
n
1
>
n
2
>
⋯
>
n
k
≥
1
1
n
i
1
σ
(
1
)
n
i
2
σ
(
2
)
⋯
n
i
k
σ
(
k
)
{\displaystyle \sum _{\sigma }\sum _{n_{1}>n_{2}>\cdots >n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}}
, and a term
1
n
1
i
1
n
2
i
2
⋯
n
k
i
k
{\displaystyle {\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}}
occurs on the left-hand since once if all the
n
i
{\displaystyle n_{i}}
are distinct, and not at all otherwise. Thus, it suffices to show
∑
Π
⪰
Λ
c
~
(
Π
)
=
{
1
,
if
|
Λ
|
=
k
0
,
otherwise
.
{\displaystyle \sum _{\Pi \succeq \Lambda }{\tilde {c}}(\Pi )={\begin{cases}1,{\text{ if }}\left|\Lambda \right|=k\\0,{\text{ otherwise }}.\end{cases}}}
(1)
To prove this, note first that the sign of
c
~
(
Π
)
{\displaystyle {\tilde {c}}(\Pi )}
is positive if the permutations of cycle-type
Π
{\displaystyle \Pi }
are even, and negative if they are odd: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group
∑
k
(
n
)
{\displaystyle \sum _{k}(n)}
. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition
Λ
{\displaystyle \Lambda }
is
{
{
1
}
,
{
2
}
,
⋯
,
{
k
}
}
{\displaystyle \{\{1\},\{2\},\cdots ,\{k\}\}}
.[ 4]
The sum and duality conjectures[ 4]
We first state the sum conjecture, which is due to C. Moen.[ 6]
Sum conjecture(Hoffman). For positive integers k=n,
∑
i
1
+
⋯
+
i
k
=
n
,
i
1
>
1
ζ
(
i
1
,
⋯
,
i
k
)
=
ζ
(
n
)
{\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}\zeta (i_{1},\cdots ,i_{k})=\zeta (n)}
, where the sum is extended over k-tuples
i
1
,
⋯
,
i
k
{\displaystyle i_{1},\cdots ,i_{k}}
of positive integers with
i
1
>
1
{\displaystyle i_{1}>1}
.
There remarks concerning this conjecture are in order. First, it implies
∑
i
1
+
⋯
+
i
k
=
n
,
i
1
>
1
S
(
i
1
,
⋯
,
i
k
)
=
(
n
−
1
k
−
1
)
ζ
(
n
)
{\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}S(i_{1},\cdots ,i_{k})={n-1 \choose k-1}\zeta (n)}
. Second, in the case
k
=
2
{\displaystyle k=2}
it says that
ζ
(
n
−
1
,
1
)
+
ζ
(
n
−
2
,
2
)
+
⋯
+
ζ
(
2
,
n
−
2
)
=
ζ
(
n
)
{\displaystyle \zeta (n-1,1)+\zeta (n-2,2)+\cdots +\zeta (2,n-2)=\zeta (n)}
, or using the relation between the
ζ
′
s
{\displaystyle \zeta 's}
and
S
′
s
{\displaystyle S's}
and Theorem 1,
2
S
(
n
−
1
,
1
)
=
(
n
+
1
)
ζ
(
n
)
−
∑
k
=
2
n
−
2
ζ
(
k
)
ζ
(
n
−
k
)
.
{\displaystyle 2S(n-1,1)=(n+1)\zeta (n)-\sum _{k=2}^{n-2}\zeta (k)\zeta (n-k).}
This was proved by Euler's paper[ 7] and has been rediscovered several times, in particular by Williams.[ 8] Finally, C. Moen[ 6] has proved the same conjecture for k=3 by lengthy but elementary arguments.
For the duality conjecture, we first define an involution
τ
{\displaystyle \tau }
on the set
ℑ
{\displaystyle \Im }
of finite sequences of positive integers whose first element is greater than 1. Let
T
{\displaystyle \mathrm {T} }
be the set of strictly increasig finite sequences of positive integers, and let
Σ
:
ℑ
→
T
{\displaystyle \Sigma :\Im \rightarrow \mathrm {T} }
be the function that sends a sequence in
ℑ
{\displaystyle \Im }
to its sequence of partial sums. If
T
n
{\displaystyle \mathrm {T} _{n}}
is the set of sequences in
T
a
u
{\displaystyle Tau}
whose last element is at most
n
{\displaystyle n}
, we have two commuting involutions
R
n
{\displaystyle R_{n}}
and
C
n
{\displaystyle C_{n}}
on
T
n
{\displaystyle \mathrm {T} _{n}}
defined by
R
n
(
a
1
,
a
2
,
⋯
,
a
l
)
=
(
n
+
1
−
a
l
,
n
+
1
−
a
l
−
1
,
⋯
,
n
+
1
−
a
1
)
{\displaystyle R_{n}(a_{1},a_{2},\cdots ,a_{l})=(n+1-a_{l},n+1-a_{l-1},\cdots ,n+1-a_{1})}
and
C
n
(
a
1
,
⋯
,
a
l
)
{\displaystyle C_{n}(a_{1},\cdots ,a_{l})}
= complement of
{
a
1
,
⋯
,
a
l
}
{\displaystyle \{a_{1},\cdots ,a_{l}\}}
in
{
1
,
2
,
⋯
,
n
}
{\displaystyle \{1,2,\cdots ,n\}}
arranged in increasing order. The our definition of
τ
{\displaystyle \tau }
is
τ
(
I
)
=
Σ
−
1
R
n
C
n
Σ
(
I
)
=
Σ
−
1
C
n
R
n
Σ
(
I
)
{\displaystyle \tau (I)=\Sigma ^{-1}R_{n}C_{n}\Sigma (I)=\Sigma ^{-1}C_{n}R_{n}\Sigma (I)}
for
I
=
(
i
1
,
i
2
,
⋯
,
i
k
)
∈
ℑ
{\displaystyle I=(i_{1},i_{2},\cdots ,i_{k})\in \Im }
with
i
1
+
⋯
+
i
k
=
n
{\displaystyle i_{1}+\cdots +i_{k}=n}
.
For example,
τ
(
3
,
4
,
1
)
=
Σ
−
1
C
8
R
8
(
3
,
7
,
8
)
=
Σ
−
1
(
3
,
4
,
5
,
7
,
8
)
=
(
3
,
1
,
1
,
2
,
1
)
.
{\displaystyle \tau (3,4,1)=\Sigma ^{-1}C_{8}R_{8}(3,7,8)=\Sigma ^{-1}(3,4,5,7,8)=(3,1,1,2,1).}
We shall say the sequences
(
i
1
,
⋯
,
i
k
)
{\displaystyle (i_{1},\cdots ,i_{k})}
and
τ
(
i
1
,
⋯
,
i
k
)
{\displaystyle \tau (i_{1},\cdots ,i_{k})}
are dual to each other, and refer to a sequence fixed by
τ
{\displaystyle \tau }
as self-dual.[ 4]
Duality conjecture (Hoffman). If
(
h
1
,
⋯
,
h
n
−
k
)
{\displaystyle (h_{1},\cdots ,h_{n-k})}
is dual to
(
i
1
,
⋯
,
i
k
)
{\displaystyle (i_{1},\cdots ,i_{k})}
, then
ζ
(
h
1
,
⋯
,
h
n
−
k
)
=
ζ
(
i
1
,
⋯
,
i
k
)
{\displaystyle \zeta (h_{1},\cdots ,h_{n-k})=\zeta (i_{1},\cdots ,i_{k})}
.
This sum conjecture is also known as Sum Theorem , and it may be expressed as follows: the Riemann zeta value of an integer n ≥ 2 is equal to the sum of all the valid (i.e. with s 1 > 1) MZVs of the partitions of length k and weight n , with 1 ≤ k ≤n − 1. In formula:[ 1]
∑
s
1
>
1
s
1
+
⋯
+
s
k
=
n
ζ
(
s
1
,
…
,
s
k
)
=
ζ
(
n
)
{\displaystyle \sum _{\stackrel {s_{1}+\cdots +s_{k}=n}{s_{1}>1}}\zeta (s_{1},\ldots ,s_{k})=\zeta (n)}
For example with length k = 2 and weight n = 7:
ζ
(
6
,
1
)
+
ζ
(
5
,
2
)
+
ζ
(
4
,
3
)
+
ζ
(
3
,
4
)
+
ζ
(
2
,
5
)
=
ζ
(
7
)
{\displaystyle \zeta (6,1)+\zeta (5,2)+\zeta (4,3)+\zeta (3,4)+\zeta (2,5)=\zeta (7)}
Euler sum with all possible alternations of sign
The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.[ 3]
Notation
∑
n
=
1
∞
H
n
(
b
)
(
−
1
)
(
n
+
1
)
(
n
+
1
)
a
=
ζ
(
a
¯
,
b
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}}=\zeta ({\bar {a}},b)}
with
H
n
(
b
)
=
+
1
+
1
2
b
+
1
3
b
+
⋯
{\displaystyle H_{n}^{(b)}=+1+{\frac {1}{2^{b}}}+{\frac {1}{3^{b}}}+\cdots }
are the generalized harmonic numbers .
∑
n
=
1
∞
H
¯
n
(
b
)
(
n
+
1
)
a
=
ζ
(
a
,
b
¯
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(b)}}{(n+1)^{a}}}=\zeta (a,{\bar {b}})}
with
H
¯
n
(
b
)
=
−
1
+
1
2
b
−
1
3
b
+
⋯
{\displaystyle {\bar {H}}_{n}^{(b)}=-1+{\frac {1}{2^{b}}}-{\frac {1}{3^{b}}}+\cdots }
∑
n
=
1
∞
H
¯
n
(
b
)
(
−
1
)
(
n
+
1
)
(
n
+
1
)
a
=
ζ
(
a
¯
,
b
¯
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}}=\zeta ({\bar {a}},{\bar {b}})}
∑
n
=
1
∞
(
−
1
)
n
(
n
+
2
)
a
∑
n
=
1
∞
H
¯
n
(
c
)
(
−
1
)
(
n
+
1
)
(
n
+
1
)
b
=
ζ
(
a
¯
,
b
¯
,
c
¯
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}}=\zeta ({\bar {a}},{\bar {b}},{\bar {c}})}
with
H
¯
n
(
c
)
=
−
1
+
1
2
c
−
1
3
c
+
⋯
{\displaystyle {\bar {H}}_{n}^{(c)}=-1+{\frac {1}{2^{c}}}-{\frac {1}{3^{c}}}+\cdots }
∑
n
=
1
∞
(
−
1
)
n
(
n
+
2
)
a
∑
n
=
1
∞
H
n
(
c
)
(
n
+
1
)
b
=
ζ
(
a
¯
,
b
,
c
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {H_{n}^{(c)}}{(n+1)^{b}}}=\zeta ({\bar {a}},b,c)}
with
H
n
(
c
)
=
+
1
+
1
2
c
+
1
3
c
+
⋯
{\displaystyle H_{n}^{(c)}=+1+{\frac {1}{2^{c}}}+{\frac {1}{3^{c}}}+\cdots }
∑
n
=
1
∞
1
(
n
+
2
)
a
∑
n
=
1
∞
H
n
(
c
)
(
−
1
)
(
n
+
1
)
(
n
+
1
)
b
=
ζ
(
a
,
b
¯
,
c
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {H_{n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}}=\zeta (a,{\bar {b}},c)}
∑
n
=
1
∞
1
(
n
+
2
)
a
∑
n
=
1
∞
H
¯
n
(
c
)
(
n
+
1
)
b
=
ζ
(
a
,
b
,
c
¯
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(c)}}{(n+1)^{b}}}=\zeta (a,b,{\bar {c}})}
As a variant of the Dirichlet eta function we define
ϕ
(
s
)
=
1
−
2
(
s
−
1
)
2
(
s
−
1
)
ζ
(
s
)
{\displaystyle \phi (s)={\frac {1-2^{(s-1)}}{2^{(s-1)}}}\zeta (s)}
with
s
>
1
{\displaystyle s>1}
ϕ
(
1
)
=
−
ln
2
{\displaystyle \phi (1)=-\ln 2}
The reflection formula
ζ
(
a
,
b
)
+
ζ
(
b
,
a
)
=
ζ
(
a
)
ζ
(
b
)
−
ζ
(
a
+
b
)
{\displaystyle \zeta (a,b)+\zeta (b,a)=\zeta (a)\zeta (b)-\zeta (a+b)}
can be generalized as follows:
ζ
(
a
,
b
¯
)
+
ζ
(
b
¯
,
a
)
=
ζ
(
a
)
ϕ
(
b
)
−
ϕ
(
a
+
b
)
{\displaystyle \zeta (a,{\bar {b}})+\zeta ({\bar {b}},a)=\zeta (a)\phi (b)-\phi (a+b)}
ζ
(
a
¯
,
b
)
+
ζ
(
b
,
a
¯
)
=
ζ
(
b
)
ϕ
(
a
)
−
ϕ
(
a
+
b
)
{\displaystyle \zeta ({\bar {a}},b)+\zeta (b,{\bar {a}})=\zeta (b)\phi (a)-\phi (a+b)}
ζ
(
a
¯
,
b
¯
)
+
ζ
(
b
¯
,
a
¯
)
=
ϕ
(
a
)
ϕ
(
b
)
−
ζ
(
a
+
b
)
{\displaystyle \zeta ({\bar {a}},{\bar {b}})+\zeta ({\bar {b}},{\bar {a}})=\phi (a)\phi (b)-\zeta (a+b)}
if
a
=
b
{\displaystyle a=b}
we have
ζ
(
a
¯
,
a
¯
)
=
1
2
[
ϕ
2
(
a
)
−
ζ
(
2
a
)
]
{\displaystyle \zeta ({\bar {a}},{\bar {a}})={\tfrac {1}{2}}{\Big [}\phi ^{2}(a)-\zeta (2a){\Big ]}}
Other relations
Using the series definition it is easy to prove:
ζ
(
a
,
b
)
+
ζ
(
a
,
b
¯
)
+
ζ
(
a
¯
,
b
)
+
ζ
(
a
¯
,
b
¯
)
=
ζ
(
a
,
b
)
2
(
a
+
b
−
2
)
{\displaystyle \zeta (a,b)+\zeta (a,{\bar {b}})+\zeta ({\bar {a}},b)+\zeta ({\bar {a}},{\bar {b}})={\frac {\zeta (a,b)}{2^{(a+b-2)}}}}
with
a
>
1
{\displaystyle a>1}
ζ
(
a
,
b
,
c
)
+
ζ
(
a
,
b
,
c
¯
)
+
ζ
(
a
,
b
¯
,
c
)
+
ζ
(
a
¯
,
b
,
c
)
+
ζ
(
a
,
b
¯
,
c
¯
)
+
ζ
(
a
¯
,
b
,
c
¯
)
+
ζ
(
a
¯
,
b
¯
,
c
)
+
ζ
(
a
¯
,
b
¯
,
c
¯
)
=
ζ
(
a
,
b
,
c
)
2
(
a
+
b
+
c
−
3
)
{\displaystyle \zeta (a,b,c)+\zeta (a,b,{\bar {c}})+\zeta (a,{\bar {b}},c)+\zeta ({\bar {a}},b,c)+\zeta (a,{\bar {b}},{\bar {c}})+\zeta ({\bar {a}},b,{\bar {c}})+\zeta ({\bar {a}},{\bar {b}},c)+\zeta ({\bar {a}},{\bar {b}},{\bar {c}})={\frac {\zeta (a,b,c)}{2^{(a+b+c-3)}}}}
with
a
>
1
{\displaystyle a>1}
A further useful relation is:[ 3]
ζ
(
a
,
b
)
+
ζ
(
a
¯
,
b
¯
)
=
∑
s
>
0
(
a
+
b
−
s
−
1
)
!
[
Z
a
(
a
+
b
−
s
,
s
)
(
a
−
s
)
!
(
b
−
1
)
!
+
Z
b
(
a
+
b
−
s
,
s
)
(
b
−
s
)
!
(
a
−
1
)
!
]
{\displaystyle \zeta (a,b)+\zeta ({\bar {a}},{\bar {b}})=\sum _{s>0}(a+b-s-1)!{\Big [}{\frac {Z_{a}(a+b-s,s)}{(a-s)!(b-1)!}}+{\frac {Z_{b}(a+b-s,s)}{(b-s)!(a-1)!}}{\Big ]}}
where
Z
a
(
s
,
t
)
=
ζ
(
s
,
t
)
+
ζ
(
s
¯
,
t
)
−
[
ζ
(
s
,
t
)
+
ζ
(
s
+
t
)
]
2
(
s
−
1
)
{\displaystyle Z_{a}(s,t)=\zeta (s,t)+\zeta ({\bar {s}},t)-{\frac {{\Big [}\zeta (s,t)+\zeta (s+t){\Big ]}}{2^{(s-1)}}}}
and
Z
b
(
s
,
t
)
=
ζ
(
s
,
t
)
2
(
s
−
1
)
{\displaystyle Z_{b}(s,t)={\frac {\zeta (s,t)}{2^{(s-1)}}}}
Note that
s
{\displaystyle s}
must be used for all value
>
1
{\displaystyle >1}
for whom the argument of the factorials is
⩾
0
{\displaystyle \geqslant 0}
Other results
For any integer positive :
a
,
b
,
…
,
k
{\displaystyle a,b,\dots ,k}
:
∑
n
=
2
∞
ζ
(
n
,
k
)
=
ζ
(
k
+
1
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,k)=\zeta (k+1)}
or more generally:
∑
n
=
2
∞
ζ
(
n
,
a
,
b
,
…
,
k
)
=
ζ
(
a
+
1
,
b
,
…
,
k
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,a,b,\dots ,k)=\zeta (a+1,b,\dots ,k)}
∑
n
=
2
∞
ζ
(
n
,
k
¯
)
=
−
ϕ
(
k
+
1
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {k}})=-\phi (k+1)}
∑
n
=
2
∞
ζ
(
n
,
a
¯
,
b
)
=
ζ
(
a
+
1
¯
,
b
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {a}},b)=\zeta ({\overline {a+1}},b)}
∑
n
=
2
∞
ζ
(
n
,
a
,
b
¯
)
=
ζ
(
a
+
1
,
b
¯
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,a,{\bar {b}})=\zeta (a+1,{\bar {b}})}
∑
n
=
2
∞
ζ
(
n
,
a
¯
,
b
¯
)
=
ζ
(
a
+
1
¯
,
b
¯
)
{\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {a}},{\bar {b}})=\zeta ({\overline {a+1}},{\bar {b}})}
lim
k
→
∞
ζ
(
n
,
k
)
=
ζ
(
n
)
−
1
{\displaystyle \lim _{k\to \infty }\zeta (n,k)=\zeta (n)-1}
1
−
ζ
(
2
)
+
ζ
(
3
)
−
ζ
(
4
)
+
⋯
=
|
1
2
|
{\displaystyle 1-\zeta (2)+\zeta (3)-\zeta (4)+\cdots =|{\frac {1}{2}}|}
ζ
(
a
,
a
)
=
1
2
[
(
ζ
(
a
)
)
2
−
ζ
(
2
a
)
]
{\displaystyle \zeta (a,a)={\tfrac {1}{2}}{\Big [}(\zeta (a))^{2}-\zeta (2a){\Big ]}}
ζ
(
a
,
a
,
a
)
=
1
6
(
ζ
(
a
)
)
3
+
1
3
ζ
(
3
a
)
−
1
2
ζ
(
a
)
ζ
(
2
a
)
{\displaystyle \zeta (a,a,a)={\tfrac {1}{6}}(\zeta (a))^{3}+{\tfrac {1}{3}}\zeta (3a)-{\tfrac {1}{2}}\zeta (a)\zeta (2a)}
Mordell–Tornheim zeta values
The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950) , is defined by
ζ
M
T
,
r
(
s
1
,
…
,
s
r
;
s
r
+
1
)
=
∑
m
1
,
…
,
m
r
>
0
1
m
1
s
1
⋯
m
r
s
r
(
m
1
+
⋯
+
m
r
)
s
r
+
1
{\displaystyle \zeta _{MT,r}(s_{1},\dots ,s_{r};s_{r+1})=\sum _{m_{1},\dots ,m_{r}>0}{\frac {1}{m_{1}^{s_{1}}\cdots m_{r}^{s_{r}}(m_{1}+\dots +m_{r})^{s_{r+1}}}}}
It is a special case of the Shintani zeta function .
References
Tornheim, Leonard (1950). "Harmonic double series". American Journal of Mathematics . 72 : 303–314. doi :10.2307/2372034 . ISSN 0002-9327 . MR 0034860 .
Mordell, Louis J. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society . Second Series. 33 : 368–371. doi :10.1112/jlms/s1-33.3.368 . ISSN 0024-6107 . MR 0100181 .
Apostol, Tom M. ; Vu, Thiennu H. (1984), "Dirichlet series related to the Riemann zeta function", Journal of Number Theory , 19 (1): 85–102, doi :10.1016/0022-314X(84)90094-5 , ISSN 0022-314X , MR 0751166
Crandall, Richard E.; Buhler, Joe P. (1994). "On the evaluation of Euler Sums" . Experimental Mathematics . 3 (4): 275. MR 1341720 .
Borwein, Jonathan M.; Girgensohn, Roland (1996). "Evaluation of Triple Euler Sums" . El. J. Combinat . 3 (1): #R23. MR 1401442 .
Flajolet, Philippe; Salvy, Bruno (1998). "Euler Sums and contour integral representations" . Exp. Math . 7 .
Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proceedings of the American Mathematical Society . 128 (5): 1275–1283. doi :10.1090/S0002-9939-99-05398-8 . MR 1670846 .
Matsumoto, Kohji (2003), "On Mordell–Tornheim and other multiple zeta-functions", Proceedings of the Session in Analytic Number Theory and Diophantine Equations , Bonner Math. Schriften, vol. 360, Bonn: Univ. Bonn, MR 2075634
Espinosa, Olivier; Moll, Victor H. (2008). "The evaluation of Tornheim double sums". arXiv :0811.0557 .
Espinosa, Olivier; Moll, Victor H. (2010). "The evaluation of Tornheim double sums II". Ramanujan J . 22 : 55–99. doi :10.1007/s11139-009-9181-1 . MR 2610609 .
Borwein, J.M. ; Chan, O-Y. (2010). "Duality in tails of multiple zeta values". Int. J. Number Theory . 6 (3): 501–514. doi :10.1142/S1793042110003058 . MR 2652893 .
Basu, Ankur (2011). "On the evaluation of Tornheim sums and allied double sums". Ramanujan J . 26 (2): 193–207. doi :10.1007/s11139-011-9302-5 . MR 2853480 .
Notes
^ a b Hoffman, Mike. "Multiple Zeta Values" . Mike Hoffman's Home Page . U.S. Naval Academy. Retrieved June 8, 2012 .
^ a b Borwein, David; Borwein, Jonathan; Bradley, David (September 23, 2004). "Parametric Euler Sum Identities" (PDF) . CARMA, AMSI Honours Course . The University of Newcastle. Retrieved June 3, 2012 .
^ a b c d Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory". arXiv :hep-th/9604128 .
^ a b c d Hoffman, Michael (1992). "Multiple Harmonic Series" . Pacific Journal of Mathematics . 152 : 276–278. MR 1141796 . Zbl 0763.11037 .
^ Ramachandra Rao, R. Sita (1984). "Transformation formulae for multiple series". Pacific Journal of Mathematics . 113 : 417–479.
^ a b Moen, C. "Sums of Simple Series". Preprint .
^ Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol . 15 (20): 140–186.
^ Williams, G. T. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society . 33 : 368–371.
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