In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by and converge when Re(s1) + ... + Re(si) > i for all i.
When s1, ..., sk are all positive integers (with s1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums.
[1][2] The k in the above definition is named the "depth" of a MZV, and the n = s1 + ... + sk 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, Multiple zeta functions arise as special cases of the multiple polylogarithms which are generalizations of the polylogarithm functions.
are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level
, they are called Euler sums or alternating multiple zeta values, and when
they are simply called multiple zeta values.
, so for example 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 Using this convention, the result can be stated as follows:[2] This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that To utilize this in the context of multiple zeta values, define
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
, and define which, by the aforementioned integral identity, makes Then, the integral identity on products gives[2] In the particular case of only two parameters we have (with s > 1 and n, m integers):[4] Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler: 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):[4] Note that if
[5] In the particular case of only three parameters we have (with a > 1 and n, j, i integers): The above MZVs satisfy the Euler reflection formula: Using the shuffle relations, it is easy to prove that:[5] This function can be seen as a generalization of the reflection formulas.
It occurs on the right-hand side in those terms corresponding to partitions
has a unique cycle type specified by a partition that refines
We follow the same line of argument as in the preceding proof.
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
But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition
[6] Source:[6] We first state the sum conjecture, which is due to C.
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.
of finite sequences of positive integers whose first element is greater than 1.
be the set of strictly increasing finite sequences of positive integers, and let
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:[3] For example, with length k = 2 and weight n = 7: The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.
[5] As a variant of the Dirichlet eta function we define The reflection formula
Using the series definition it is easy to prove: A further useful relation is:[5] where
: The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950), is defined by It is a special case of the Shintani zeta function.