Multiple zeta function

Two parameters case

In the particular case of only two parameters we have (with s>1 and n,m integer):

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 ζ ( s , t ) = ∑ n = 1 ∞ H n , t ( n + 1 ) s where 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 ,

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):

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 )

Note that if s + t = 2 p + 2 we have p / 3 irreducibles, i.e. these MZVs cannot be written as function of ζ ( a ) only.

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

Euler reflection formula

The above MZVs satisfy the Euler reflection formula:

ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) − ζ ( a + b ) for 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 ) for 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 , and for a partition Π = { P 1 , P 2 , … , P l } of the set { 1 , 2 , … , k } , let c ( Π ) = ( | P 1 | − 1 ) ! ( | P 2 | − 1 ) ! ⋯ ( | P l | − 1 ) ! . Also, given such a Π and a k-tuple i = { i 1 , . . . , i k } of exponents, define ∏ s = 1 l ζ ( ∑ j ∈ P s i j ) .

The relations between the ζ and S are: S ( i 1 , i 2 ) = ζ ( i 1 , i 2 ) + ζ ( 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 )

Theorem 1(Hoffman)

For any real i 1 , ⋯ , i k > 1 , , ∑ σ ∈ ∑ k S ( i σ ( 1 ) , … , i σ ( k ) ) = ∑ partitions  Π  of  { 1 , … , k } c ( Π ) ζ ( i , Π ) .

Proof. Assume the 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 ) . Now thinking on the symmetric

group ∑ k as acting on k-tuple n = ( 1 , ⋯ , k ) of positive integers. A given k-tuple n = ( n 1 , ⋯ , n k ) has an isotropy group

∑ k ( n ) and an associated partition Λ of ( 1 , 2 , ⋯ , k ) : Λ is the set of equivalence classes of the relation given by i ∼ j iff n i = n j , and ∑ k ( n ) = { σ ∈ ∑ k : σ ( i ) ∼ ∀ i } . Now the term 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( 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 ) exactly | ∑ k ( n ) | times. It occurs on the right-hand side in those terms corresponding to partitions Π that are refinements of Λ : letting ⪰ denote refinement, 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( k ) occurs ∑ Π ⪰ Λ ( Π ) times. Thus, the conclusion will follow if | ∑ k ( n ) | = ∑ Π ⪰ Λ c ( Π ) for any k-tuple n = { n 1 , ⋯ , n k } and associated partition Λ . To see this, note that c ( Π ) counts the permutations having cycle-type specified by Π : since any elements of ∑ k ( n ) has a unique cycle-type specified by a partition that refines Λ , the result follows.

For 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 ) for 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 . To state the analog of Theorem 1 for the ζ ′ s , we require one bit of notation. For a partition

Π = { P 1 , ⋯ , P l } or { 1 , 2 ⋯ , k } , let c ~ ( Π ) = ( − 1 ) k − l c ( Π ) .

Theorem 2(Hoffman)

For any real i 1 , ⋯ , i k > 1 , ∑ σ ∈ ∑ k ζ ( i σ ( 1 ) , … , i σ ( k ) ) = ∑ partitions  Π  of  { 1 , … , k } c ~ ( Π ) ζ ( i , Π ) .

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 ) , and a term 1 n 1 i 1 n 2 i 2 ⋯ n k i k occurs on the left-hand since once if all the n i are distinct, and not at all otherwise. Thus, it suffices to show ∑ Π ⪰ Λ c ~ ( Π ) = { 1 ,  if  | Λ | = k 0 ,  otherwise  . (1)

To prove this, note first that the sign of c ~ ( Π ) is positive if the permutations of cycle-type Π 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 ) . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition Λ is { { 1 } , { 2 } , ⋯ , { 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 ) , where the sum is extended over k-tuples i 1 , ⋯ , i k of positive integers with 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 ) . Second, in the case k = 2 it says that ζ ( n − 1 , 1 ) + ζ ( n − 2 , 2 ) + ⋯ + ζ ( 2 , n − 2 ) = ζ ( n ) , or using the relation between the ζ ′ s and S ′ s and Theorem 1, 2 S ( n − 1 , 1 ) = ( n + 1 ) ζ ( n ) − ∑ k = 2 n − 2 ζ ( k ) ζ ( n − k ) .

This was proved by Euler's paper 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 on the set ℑ of finite sequences of positive integers whose first element is greater than 1. Let T be the set of strictly increasing finite sequences of positive integers, and let Σ : ℑ → T be the function that sends a sequence in ℑ to its sequence of partial sums. If T n is the set of sequences in T a u whose last element is at most n , we have two commuting involutions R n and C n on 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 ) and C n ( a 1 , ⋯ , a l ) = complement of { a 1 , ⋯ , a l } in { 1 , 2 , ⋯ , n } arranged in increasing order. The our definition of τ is τ ( I ) = Σ − 1 R n C n Σ ( I ) = Σ − 1 C n R n Σ ( I ) for I = ( i 1 , i 2 , ⋯ , i k ) ∈ ℑ with i 1 + ⋯ + 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 ) . We shall say the sequences ( i 1 , ⋯ , i k ) and τ ( i 1 , ⋯ , i k ) are dual to each other, and refer to a sequence fixed by τ as self-dual.

Duality conjecture (Hoffman). If ( h 1 , ⋯ , h n − k ) is dual to ( i 1 , ⋯ , i k ) , then ζ ( h 1 , ⋯ , h n − k ) = ζ ( i 1 , ⋯ , 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 )

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 )

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 ) with H n ( b ) = + 1 + 1 2 b + 1 3 b + ⋯ are the generalized harmonic numbers. ∑ n = 1 ∞ H ¯ n ( b ) ( n + 1 ) a = ζ ( a , b ¯ ) with H ¯ n ( b ) = − 1 + 1 2 b − 1 3 b + ⋯ ∑ n = 1 ∞ H ¯ n ( b ) ( − 1 ) ( n + 1 ) ( n + 1 ) a = ζ ( a ¯ , b ¯ ) ∑ n = 1 ∞ ( − 1 ) n ( n + 2 ) a ∑ n = 1 ∞ H ¯ n ( c ) ( − 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a ¯ , b ¯ , c ¯ ) with H ¯ n ( c ) = − 1 + 1 2 c − 1 3 c + ⋯ ∑ n = 1 ∞ ( − 1 ) n ( n + 2 ) a ∑ n = 1 ∞ H n ( c ) ( n + 1 ) b = ζ ( a ¯ , b , c ) with H n ( c ) = + 1 + 1 2 c + 1 3 c + ⋯ ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ n = 1 ∞ H n ( c ) ( − 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a , b ¯ , c ) ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ n = 1 ∞ H ¯ n ( c ) ( n + 1 ) b = ζ ( a , b , c ¯ )

As a variant of the Dirichlet eta function we define

ϕ ( s ) = 1 − 2 ( s − 1 ) 2 ( s − 1 ) ζ ( s ) with s > 1 ϕ ( 1 ) = − ln ⁡ 2

Reflection formula

The reflection formula ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) − ζ ( a + b ) can be generalized as follows:

ζ ( a , b ¯ ) + ζ ( b ¯ , a ) = ζ ( a ) ϕ ( b ) − ϕ ( a + b ) ζ ( a ¯ , b ) + ζ ( b , a ¯ ) = ζ ( b ) ϕ ( a ) − ϕ ( a + b ) ζ ( a ¯ , b ¯ ) + ζ ( b ¯ , a ¯ ) = ϕ ( a ) ϕ ( b ) − ζ ( a + b )

if a = b we have ζ ( a ¯ , a ¯ ) = 1 2 [ ϕ 2 ( a ) − ζ ( 2 a ) ]

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 ) with 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 ) with 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 ) ! ]

where Z a ( s , t ) = ζ ( s , t ) + ζ ( s ¯ , t ) − [ ζ ( s , t ) + ζ ( s + t ) ] 2 ( s − 1 ) and Z b ( s , t ) = ζ ( s , t ) 2 ( s − 1 )

Note that s must be used for all value > 1 for whom the argument of the factorials is ⩾ 0

Other results

For any integer positive : a , b , … , k :

∑ n = 2 ∞ ζ ( n , k ) = ζ ( k + 1 ) or more generally: ∑ n = 2 ∞ ζ ( n , a , b , … , k ) = ζ ( a + 1 , b , … , k ) ∑ n = 2 ∞ ζ ( n , k ¯ ) = − ϕ ( k + 1 ) ∑ n = 2 ∞ ζ ( n , a ¯ , b ) = ζ ( a + 1 ¯ , b ) ∑ n = 2 ∞ ζ ( n , a , b ¯ ) = ζ ( a + 1 , b ¯ ) ∑ n = 2 ∞ ζ ( n , a ¯ , b ¯ ) = ζ ( a + 1 ¯ , b ¯ ) lim k → ∞ ζ ( n , k ) = ζ ( n ) − 1 1 − ζ ( 2 ) + ζ ( 3 ) − ζ ( 4 ) + ⋯ = | 1 2 | ζ ( a , a ) = 1 2 [ ( ζ ( a ) ) 2 − ζ ( 2 a ) ] ζ ( a , a , a ) = 1 6 ( ζ ( a ) ) 3 + 1 3 ζ ( 3 a ) − 1 2 ζ ( a ) ζ ( 2 a )

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

It is a special case of the Shintani zeta function.