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- Source: Multiple zeta function
In mathematics, the multiple zeta functions are generalizations 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(s1) + ... + Re(si) > 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 s1, ..., sk are all positive integers (with s1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms.
The k in the above definition is named the "depth" of a MZV, and the n = s1 + ... + sk is known as the "weight".
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).}
Definition
Multiple zeta functions arise as special cases of the multiple polylogarithms
L
i
s
1
,
…
,
s
d
(
μ
1
,
…
,
μ
d
)
=
∑
k
1
>
⋯
>
k
d
>
0
μ
1
k
1
⋯
μ
d
k
d
k
1
s
1
⋯
k
d
s
d
{\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\mu _{1}^{k_{1}}\cdots \mu _{d}^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}
which are generalizations of the polylogarithm functions. When all of the
μ
i
{\displaystyle \mu _{i}}
are nth roots of unity and the
s
i
{\displaystyle s_{i}}
are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level
n
{\displaystyle n}
. In particular, when
n
=
2
{\displaystyle n=2}
, they are called Euler sums or alternating multiple zeta values, and when
n
=
1
{\displaystyle n=1}
they are simply called multiple zeta values. Multiple zeta values are often written
ζ
(
s
1
,
…
,
s
d
)
=
∑
k
1
>
⋯
>
k
d
>
0
1
k
1
s
1
⋯
k
d
s
d
{\displaystyle \zeta (s_{1},\ldots ,s_{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {1}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}
and Euler sums are written
ζ
(
s
1
,
…
,
s
d
;
ε
1
,
…
,
ε
d
)
=
∑
k
1
>
⋯
>
k
d
>
0
ε
1
k
1
⋯
ε
k
d
k
1
s
1
⋯
k
d
s
d
{\displaystyle \zeta (s_{1},\ldots ,s_{d};\varepsilon _{1},\ldots ,\varepsilon _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\varepsilon _{1}^{k_{1}}\cdots \varepsilon ^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}}
where
ε
i
=
±
1
{\displaystyle \varepsilon _{i}=\pm 1}
. Sometimes, authors will write a bar over an
s
i
{\displaystyle s_{i}}
corresponding to an
ε
i
{\displaystyle \varepsilon _{i}}
equal to
−
1
{\displaystyle -1}
, so for example
ζ
(
a
¯
,
b
)
=
ζ
(
a
,
b
;
−
1
,
1
)
{\displaystyle \zeta ({\overline {a}},b)=\zeta (a,b;-1,1)}
.
Integral structure and identities
It was noticed by Kontsevich that it is possible to express colored multiple zeta values (and thus their special cases) as certain multivariable integrals. This result is often stated with the use of a convention for iterated integrals, wherein
∫
0
x
f
1
(
t
)
d
t
⋯
f
d
(
t
)
d
t
=
∫
0
x
f
1
(
t
1
)
(
∫
0
t
1
f
2
(
t
2
)
(
∫
0
t
2
⋯
(
∫
0
t
d
f
d
(
t
d
)
d
t
d
)
)
d
t
2
)
d
t
1
{\displaystyle \int _{0}^{x}f_{1}(t)dt\cdots f_{d}(t)dt=\int _{0}^{x}f_{1}(t_{1})\left(\int _{0}^{t_{1}}f_{2}(t_{2})\left(\int _{0}^{t_{2}}\cdots \left(\int _{0}^{t_{d}}f_{d}(t_{d})dt_{d}\right)\right)dt_{2}\right)dt_{1}}
Using this convention, the result can be stated as follows:
L
i
s
1
,
…
,
s
d
(
μ
1
,
…
,
μ
d
)
=
∫
0
1
(
d
t
t
)
s
1
−
1
d
t
a
1
−
t
⋯
(
d
t
t
)
s
d
−
1
d
t
a
d
−
t
{\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\int _{0}^{1}\left({\frac {dt}{t}}\right)^{s_{1}-1}{\frac {dt}{a_{1}-t}}\cdots \left({\frac {dt}{t}}\right)^{s_{d}-1}{\frac {dt}{a_{d}-t}}}
where
a
j
=
∏
i
=
1
j
μ
i
−
1
{\displaystyle a_{j}=\prod \limits _{i=1}^{j}\mu _{i}^{-1}}
for
j
=
1
,
2
,
…
,
d
{\displaystyle j=1,2,\ldots ,d}
.
This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that
(
∫
0
x
f
1
(
t
)
d
t
⋯
f
n
(
t
)
d
t
)
(
∫
0
x
f
n
+
1
(
t
)
d
t
⋯
f
m
(
t
)
d
t
)
=
∑
σ
∈
S
h
n
,
m
∫
0
x
f
σ
(
1
)
(
t
)
⋯
f
σ
(
m
)
(
t
)
{\displaystyle \left(\int _{0}^{x}f_{1}(t)dt\cdots f_{n}(t)dt\right)\!\left(\int _{0}^{x}f_{n+1}(t)dt\cdots f_{m}(t)dt\right)=\sum \limits _{\sigma \in {\mathfrak {Sh}}_{n,m}}\int _{0}^{x}f_{\sigma (1)}(t)\cdots f_{\sigma (m)}(t)}
where
S
h
n
,
m
=
{
σ
∈
S
m
∣
σ
(
1
)
<
⋯
<
σ
(
n
)
,
σ
(
n
+
1
)
<
⋯
<
σ
(
m
)
}
{\displaystyle {\mathfrak {Sh}}_{n,m}=\{\sigma \in S_{m}\mid \sigma (1)<\cdots <\sigma (n),\sigma (n+1)<\cdots <\sigma (m)\}}
and
S
m
{\displaystyle S_{m}}
is the symmetric group on
m
{\displaystyle m}
symbols.
To utilize this in the context of multiple zeta values, define
X
=
{
a
,
b
}
{\displaystyle X=\{a,b\}}
,
X
∗
{\displaystyle X^{*}}
to be the free monoid generated by
X
{\displaystyle X}
and
A
{\displaystyle {\mathfrak {A}}}
to be the free
Q
{\displaystyle \mathbb {Q} }
-vector space generated by
X
∗
{\displaystyle X^{*}}
.
A
{\displaystyle {\mathfrak {A}}}
can be equipped with the shuffle product, turning it into an algebra. Then, the multiple zeta function can be viewed as an evaluation map, where we identify
a
=
d
t
t
{\displaystyle a={\frac {dt}{t}}}
,
b
=
d
t
1
−
t
{\displaystyle b={\frac {dt}{1-t}}}
, and define
ζ
(
w
)
=
∫
0
1
w
{\displaystyle \zeta (\mathbf {w} )=\int _{0}^{1}\mathbf {w} }
for any
w
∈
X
∗
{\displaystyle \mathbf {w} \in X^{*}}
,
which, by the aforementioned integral identity, makes
ζ
(
a
s
1
−
1
b
⋯
a
s
d
−
1
b
)
=
ζ
(
s
1
,
…
,
s
d
)
.
{\displaystyle \zeta (a^{s_{1}-1}b\cdots a^{s_{d}-1}b)=\zeta (s_{1},\ldots ,s_{d}).}
Then, the integral identity on products gives
ζ
(
w
)
ζ
(
v
)
=
ζ
(
w
⧢
v
)
.
{\displaystyle \zeta (w)\zeta (v)=\zeta (w{\text{ ⧢ }}v).}
Two parameters case
In the particular case of only two parameters we have (with s > 1 and n, m integers):
ζ
(
s
,
t
)
=
∑
n
>
m
≥
1
1
n
s
m
t
=
∑
n
=
2
∞
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=2}^{\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 Hn 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):
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)}
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.
Three parameters case
In the particular case of only three parameters we have (with a > 1 and n, j, i integers):
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
j
,
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_{j,c}}{(j+1)^{b}}}}
Euler reflection formula
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:
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
∈
Σ
k
S
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \Sigma _{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 no 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
Σ
k
{\displaystyle \Sigma _{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
Σ
k
(
n
)
{\displaystyle \Sigma _{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
Σ
k
(
n
)
=
{
σ
∈
Σ
k
:
σ
(
i
)
∼
∀
i
}
{\displaystyle \Sigma _{k}(n)=\{\sigma \in \Sigma _{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
∈
Σ
k
S
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \Sigma _{k}}S(i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}c(\Pi )\zeta (i,\Pi )}
exactly
|
Σ
k
(
n
)
|
{\displaystyle \left|\Sigma _{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
|
Σ
k
(
n
)
|
=
∑
Π
⪰
Λ
c
(
Π
)
{\displaystyle \left|\Sigma _{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
Σ
k
(
n
)
{\displaystyle \Sigma _{k}(n)}
has a unique cycle type specified by a partition that refines
Λ
{\displaystyle \Lambda }
, the result follows.
For
k
=
3
{\displaystyle k=3}
, the theorem says
∑
σ
∈
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 \Sigma _{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.
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}\}}
of
{
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
∈
Σ
k
ζ
(
i
σ
(
1
)
,
…
,
i
σ
(
k
)
)
=
∑
partitions
Π
of
{
1
,
…
,
k
}
c
~
(
Π
)
ζ
(
i
,
Π
)
{\displaystyle \sum _{\sigma \in \Sigma _{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
Σ
k
(
n
)
{\displaystyle \Sigma _{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\}\}}
.
The sum and duality conjectures
We first state the sum conjecture, which is due to C. Moen.
Sum conjecture (Hoffman). For positive integers k and 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}
.
Three 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 and has been rediscovered several times, in particular by Williams. Finally, C. Moen 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 increasing 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
{\displaystyle \mathrm {T} }
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},\dots ,a_{l})=(n+1-a_{l},n+1-a_{l-1},\dots ,n+1-a_{1})}
and
C
n
(
a
1
,
…
,
a
l
)
{\displaystyle C_{n}(a_{1},\dots ,a_{l})}
= complement of
{
a
1
,
…
,
a
l
}
{\displaystyle \{a_{1},\dots ,a_{l}\}}
in
{
1
,
2
,
…
,
n
}
{\displaystyle \{1,2,\dots ,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},\dots ,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},\dots ,i_{k})}
and
τ
(
i
1
,
…
,
i
k
)
{\displaystyle \tau (i_{1},\dots ,i_{k})}
are dual to each other, and refer to a sequence fixed by
τ
{\displaystyle \tau }
as self-dual.
Duality conjecture (Hoffman). If
(
h
1
,
…
,
h
n
−
k
)
{\displaystyle (h_{1},\dots ,h_{n-k})}
is dual to
(
i
1
,
…
,
i
k
)
{\displaystyle (i_{1},\dots ,i_{k})}
, then
ζ
(
h
1
,
…
,
h
n
−
k
)
=
ζ
(
i
1
,
…
,
i
k
)
{\displaystyle \zeta (h_{1},\dots ,h_{n-k})=\zeta (i_{1},\dots ,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 s1 > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula:
∑
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.
= 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}
= Reflection formula
=
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:
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 which the argument of the factorials is
⩾
0
{\displaystyle \geqslant 0}
Other results
For all positive integers
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 (2): 303–314. doi:10.2307/2372034. ISSN 0002-9327. JSTOR 2372034. MR 0034860.
Mordell, Louis J. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. Second Series. 33 (3): 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. doi:10.1080/10586458.1994.10504297. MR 1341720.
Borwein, Jonathan M.; Girgensohn, Roland (1996). "Evaluation of Triple Euler Sums". Electron. J. Comb. 3 (1): #R23. doi:10.37236/1247. hdl:1959.13/940394. MR 1401442.
Flajolet, Philippe; Salvy, Bruno (1998). "Euler Sums and contour integral representations". Exp. Math. 7: 15–35. CiteSeerX 10.1.1.37.652. doi:10.1080/10586458.1998.10504356.
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 Hugo (2008). "The evaluation of Tornheim double sums". arXiv:math/0505647.
Espinosa, Olivier; Moll, Victor Hugo (2010). "The evaluation of Tornheim double sums II". Ramanujan J. 22: 55–99. arXiv:0811.0557. doi:10.1007/s11139-009-9181-1. MR 2610609. S2CID 17055581.
Borwein, J.M.; Chan, O-Y. (2010). "Duality in tails of multiple zeta values". Int. J. Number Theory. 6 (3): 501–514. CiteSeerX 10.1.1.157.9158. 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. S2CID 120229489.
Notes
External links
Borwein, Jonathan; Zudilin, Wadim. "Lecture notes on the Multiple Zeta Function".
Hoffman, Michael (2012). "Multiple zeta values".
Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. ISBN 978-981-4689-39-7.
Burgos Gil, José Ignacio; Fresán, Javier. "Multiple zeta values: from numbers to motives" (PDF).
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