Multiple zeta function Last updated February 13, 2025 Generalizations of the Riemann 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(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 . These values can also be regarded as special values of the multiple polylogarithms. [ 1] [ 2]
The k in the above definition is named the "depth" of a MZV, and the n = s 1 + ... + s k is known as the "weight". [ 3]
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 n th 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: [ 2]
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 [ 2]
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): [ 4]
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 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): [ 4]
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)} s t approximate value explicit formulae OEIS 2 2 0.811742425283353643637002772406 3 4 ζ ( 4 ) {\displaystyle {\tfrac {3}{4}}\zeta (4)} A197110 3 2 0.228810397603353759768746148942 3 ζ ( 2 ) ζ ( 3 ) − 11 2 ζ ( 5 ) {\displaystyle 3\zeta (2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)} A258983 4 2 0.088483382454368714294327839086 ( ζ ( 3 ) ) 2 − 4 3 ζ ( 6 ) {\displaystyle \left(\zeta (3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)} A258984 5 2 0.038575124342753255505925464373 5 ζ ( 2 ) ζ ( 5 ) + 2 ζ ( 3 ) ζ ( 4 ) − 11 ζ ( 7 ) {\displaystyle 5\zeta (2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)} A258985 6 2 0.017819740416835988362659530248 A258947 2 3 0.711566197550572432096973806086 9 2 ζ ( 5 ) − 2 ζ ( 2 ) ζ ( 3 ) {\displaystyle {\tfrac {9}{2}}\zeta (5)-2\zeta (2)\zeta (3)} A258986 3 3 0.213798868224592547099583574508 1 2 ( ( ζ ( 3 ) ) 2 − ζ ( 6 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (3)\right)^{2}-\zeta (6)\right)} A258987 4 3 0.085159822534833651406806018872 17 ζ ( 7 ) − 10 ζ ( 2 ) ζ ( 5 ) {\displaystyle 17\zeta (7)-10\zeta (2)\zeta (5)} A258988 5 3 0.037707672984847544011304782294 5 ζ ( 3 ) ζ ( 5 ) − 147 24 ζ ( 8 ) − 5 2 ζ ( 6 , 2 ) {\displaystyle 5\zeta (3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)} A258982 2 4 0.674523914033968140491560608257 25 12 ζ ( 6 ) − ( ζ ( 3 ) ) 2 {\displaystyle {\tfrac {25}{12}}\zeta (6)-\left(\zeta (3)\right)^{2}} A258989 3 4 0.207505014615732095907807605495 10 ζ ( 2 ) ζ ( 5 ) + ζ ( 3 ) ζ ( 4 ) − 18 ζ ( 7 ) {\displaystyle 10\zeta (2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)} A258990 4 4 0.083673113016495361614890436542 1 2 ( ( ζ ( 4 ) ) 2 − ζ ( 8 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (4)\right)^{2}-\zeta (8)\right)} A258991
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. [ 5]
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}}}} 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: [ 5]
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. [ 6]
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. [ 7]
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\}\}} . [ 6]
The sum and duality conjectures Source: [ 6]
We first state the sum conjecture, which is due to C. Moen. [ 8]
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 [ 9] and has been rediscovered several times, in particular by Williams. [ 10] Finally, C. Moen [ 8] 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. [ 6]
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 s 1 > 1) MZVs of the partitions of length k and weight n , with 1 ≤ k ≤ n − 1. In formula: [ 3]
∑ 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. [ 5]
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: [ 5]
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 valuesThe 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 .
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