Generalizations of the Riemann zeta function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by
formula_1
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.
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".
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,
 formula_2
Definition.
Multiple zeta functions arise as special cases of the multiple polylogarithms
formula_3
which are generalizations of the polylogarithm functions. When all of the formula_4 are "n"th roots of unity and the formula_5 are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level formula_6. In particular, when formula_7, they are called Euler sums or alternating multiple zeta values, and when formula_8 they are simply called multiple zeta values. Multiple zeta values are often written
formula_9
and Euler sums are written
formula_10
where formula_11. Sometimes, authors will write a bar over an formula_5 corresponding to an formula_13 equal to formula_14, so for example
formula_15.
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
formula_16
Using this convention, the result can be stated as follows:
formula_17 where formula_18 for formula_19.
This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that
formula_20 where formula_21 and formula_22 is the symmetric group on formula_23 symbols.
To utilize this in the context of multiple zeta values, define formula_24, formula_25 to be the free monoid generated by formula_26 and formula_27 to be the free formula_28-vector space generated by formula_25. formula_27 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 formula_31, formula_32, and define
formula_33 for any formula_34,
which, by the aforementioned integral identity, makes
formula_35
Then, the integral identity on products gives
formula_36
Two parameters case.
In the particular case of only two parameters we have (with "s" ; 1 and "n", "m" integers):
formula_37
formula_38 where formula_39 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:
formula_40
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" = 2"N"+1 (taking if necessary "ζ"(0) = 0):
formula_41
Note that if formula_42 we have formula_43 irreducibles, i.e. these MZVs cannot be written as function of formula_44 only.
Three parameters case.
In the particular case of only three parameters we have (with "a" ; 1 and "n", "j", "i" integers):
formula_45
Euler reflection formula.
The above MZVs satisfy the Euler reflection formula:
formula_46 for formula_47
Using the shuffle relations, it is easy to prove that:
formula_48 for formula_49
This function can be seen as a generalization of the reflection formulas.
Symmetric sums in terms of the zeta function.
Let formula_50, and for a partition formula_51 of the set formula_52, let formula_53. Also, given such a formula_54 and a "k"-tuple formula_55 of exponents, define formula_56.
The relations between the formula_57 and formula_58 are:
formula_59 and formula_60
Theorem 1 (Hoffman).
For any real formula_61, formula_62.
Proof. Assume the formula_63 are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as
formula_64. Now thinking on the symmetric
group formula_65 as acting on "k"-tuple formula_66 of positive integers. A given "k"-tuple formula_67 has an isotropy group
formula_68 and an associated partition formula_69 of formula_70: formula_69 is the set of equivalence classes of the relation 
given by formula_72 iff formula_73, and formula_74. Now the term formula_75 occurs on the left-hand side of formula_76 exactly formula_77 times. It occurs on the right-hand side in those terms corresponding to partitions formula_54 that are refinements of formula_69: letting formula_80 denote refinement, formula_75 occurs formula_82 times. Thus, the conclusion will follow if 
formula_83 for any "k"-tuple formula_84 and associated partition formula_69.
To see this, note that formula_86 counts the permutations having cycle type specified by formula_54: since any elements of formula_68 has a unique cycle type specified by a partition that refines formula_69, the result follows.
For formula_90, the theorem says formula_91
for formula_92. This is the main result of.
Having formula_93. To state the analog of Theorem 1 for the formula_94, we require one bit of notation. For a partition
formula_95 of formula_96, let formula_97.
Theorem 2 (Hoffman).
For any real formula_98, formula_99.
Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now
formula_100, and a term formula_101 occurs on the left-hand since once if all the formula_102 are distinct, and not at all otherwise. Thus, it suffices to show 
formula_103 (1)
To prove this, note first that the sign of formula_104 is positive if the permutations of cycle type formula_54 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 formula_68. But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition formula_69 is 
formula_108.
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",
formula_109, where the sum is extended over "k"-tuples formula_110 of positive integers with formula_111.
Three remarks concerning this conjecture are in order. First, it implies
formula_112. Second, in the case formula_113 it says that formula_114, or using the relation between the formula_94 and formula_116 and Theorem 1, formula_117
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 formula_118 on the set formula_119 of finite sequences of positive integers whose first element is greater than 1. Let formula_120 be the set of strictly increasing finite sequences of positive integers, and let formula_121 be the function that sends a sequence in formula_119 to its sequence of partial sums. If formula_123 is the set of sequences in formula_120 whose last element is at most formula_6, we have two commuting involutions formula_126 and formula_127 on formula_123 defined by 
formula_129 and 
formula_130 = complement of formula_131 in formula_132 arranged in increasing order. The our definition of formula_118 is formula_134 for formula_135 with formula_136.
For example,
formula_137
We shall say the sequences formula_138 and formula_139 are dual to each other, and refer to a sequence fixed by formula_118 as self-dual.
Duality conjecture (Hoffman). If formula_141 is dual to formula_138, then formula_143.
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:
formula_144
For example, with length "k" = 2 and weight "n" = 7:
formula_145
Euler sum with all possible alternations of sign.
The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.
formula_146 with formula_147 are the generalized harmonic numbers.
formula_148 with formula_149
formula_150
formula_151 with formula_152
formula_153 with formula_154
formula_155
formula_156
Notation.
As a variant of the Dirichlet eta function we define
formula_157 with formula_158
formula_159
Reflection formula.
The reflection formula formula_46 can be generalized as follows:
formula_161
formula_162
formula_163
if formula_164 we have formula_165
Other relations.
Using the series definition it is easy to prove:
formula_166 with formula_167
formula_168 with formula_167
A further useful relation is: 
formula_170
where formula_171 and formula_172
Note that formula_173 must be used for all value formula_174 for which the argument of the factorials is formula_175
Other results.
For all positive integers formula_176:
formula_177 or more generally:
formula_178
formula_179
formula_180 
formula_181 
formula_182
formula_183
formula_184
formula_185
formula_186
Mordell–Tornheim zeta values.
The Mordell–Tornheim zeta function, introduced by who was motivated by the papers and , is defined by 
formula_187
It is a special case of the Shintani zeta function.
Notes.
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