APPENDIX: SPECIAL VALUES AND FUNCTIONAL

EQUATIONS OF POLYLOGARITHMS

Don Zagier

0. Introduction 1 1. The basic algebraic relation and the definition of A_{m}(F) 2 2. Examples of dilogarithm relations 6 3. Examples for higher order polylogarithms 8 4. Examples: ladders 10 5. Existence of relations among polylogarithm values of arbitrarily high order 12 6. A conjecture on linear independence 13 7. Functional equations 14 References 20

0. Introduction. In this appendix we want to give a unified explanation of the many linear relations which exist among values of polylogarithms at algebraic arguments, as well as a framework for understanding the functional equations of these functions. We will content ourselves with describing the (partially conjectural, partly proved) results, without explaining the reasons which motivate the form the answer takes. This motivation, as well as many more details of the theoretical setup, can be found in [Z], of which the present appendix (except for §7, which contains some new results) is essentially just a summary.

The classical theorem of number theory in which the “monologarithm” function Li_{1}(x) = − log(1 − x) appears is Dirichlet’s theorem, which in a weakened form says: Let A_{1}(F) denote the group of units of an algebraic number field F of degree n = r_{1} + 2r_{2} in the usual notation, D_{F} the discriminant of F. Then

(i) The map A_{1}(F) → ℝ^{n} which sends ε to the n-tuple of values of log |ε^{σ}|, σ ranging over the embeddings of F into ℂ, maps A_{1}(F) to a lattice in an (r_{1} +r_{2} −1)-dimensional subspace of ℝ^{n}; and (ii) The covolume of (= volume of a fundamental parallelogram for) this lattice is a rational multiple of |D_{F} |^{1/2}π^{−r_{2}} times the residue at s = 1 of ζ_{F} (s), the Dedekind zeta-function of F.

Our goal is to define similarly for each m > 1 groups A_{m}(F) such that (i) the mth polylogarithm function defines a map from A_{m}(F) to a lattice in ℝ^{n}, and (ii) the covolume of this lattice is up to a simple factor equal to ζ_{F} (m). Assertion (i) implies—since the lattice is finite-dimensional—a large number of relations among the polylogarithms with values in F, including all known, and conjecturally all, such relations, while (ii) implies that ζ_{F} (m) for any number field F and integer m > 1 can be expressed in finite terms in terms of values of polylogarithms with values in F. (It was in fact this latter statement, not the existence of relations among polylogarithm values, which was the original motivation for the conjectures in [Z].)

The definition of A_{m}(F) and more details of the above picture are given in §1. Several examples in the case of the dilogarithm are discussed in §2, and for higher polylogarithms in §3. The reader may want to look at these sections before reading §1, which is more theoretical. The relation between the concepts explained here and Lewin’s ladders is discussed in §4. The following section proves in the case F = ℚ that there are enough elements of A_{m}(F) to produce non-trivial relations, and in particular that there are non-trivial relations among values of Li_{m} with rational arguments for arbitrary large values of m. In §6 we state the conjecture that all relations among values of polylogarithms at algebraic arguments come from specializing functional equations (i.e., are “analytically derivable” in Lewin’s terminology) and hence—since the processes of verifying or specializing any functional equation are mechanical—in some sense trivial; of course, this is a very non-trivial use of the work “trivial,” since finding the functional equations needed is not at all an easy matter. Finally, in §7 we discuss how the same setup as is needed to understand relations among special values of polylogarithms also lets one search efficiently for functional equations, and report on some new functional equations found by H. Gangl, including his spectacular discovery of a functional equation for the hexalogarithm, the first progress beyond m = 5 in 150 years.

1. The basic algebraic relation and the definition of A_{m}(F). We are interested in looking for linear dependences of polylogarithm values at algebraic arguments. It turns out that the only combinations which work are those whose arguments satisfy two conditions. The first condition, which is the basic one, is purely algebraic. We will call collections of numbers satisfying this condition good, or more precisely (since the condition depends on m and becomes more restrictive as m grows) good at level m. For the problem of finding functional equations, discussed in §7, only the algebraic condition is needed, so the whole problem is to find good combinations of functions of one or several variables at various levels m. The second condition is inductive. At level 2, any “good” combination of arguments is mapped by the dilogarithm into a certain lattice in ℝ^{n}, as sketched in the introductory paragraphs and explained in more detail below, so the group A_{2}(F) consists of all good combinations and we automatically get non-trivial relations among dilogarithm values whenever the rank of this group is bigger than the dimension of the lattice to which it maps. (Actually, its rank is always infinite, as we will see in §5, so it is merely a question of computation to get as many relations as desired.) In higher levels m, any good combination of arguments has associated to it other combinations which are good at some smaller level m′, 2 ≤ m′ < m. Starting at the bottom (m′ = 2), we require that the image of each of these lower-level combinations under the m′-th polylogarithm map is 0 in the relevant lattice. The group A_{m}(F), defined as the set of all combinations of elements of F which satisfy this property, then is mapped by the mth polylogarithm map into a certain lattice of finite rank, so we automatically get relations among polylogarithm values if the rank of A_{m}(F) (which, again, turns out actually to be infinite) is larger than that of the lattice.

We now give the precise definitions. Let F be any field. By a “combination of elements of F ” we will mean a formal linear combination ξ = ∑_{i} n_{i} [x_{i}] where i runs over a finite

index set, the x_{i} are elements of F^{×}, and the coefficients n_{i} are integers. (Actually, all our considerations are up to torsion only, so we could allow n_{i} ∈ ℚ.) For convenience we also allow [x_{i}] to be 0 or ∞ but set [0] = [∞] = 0. The set of all combinations ξ forms a group F_{F}, the free abelian group on F^{×}. An element ξ ∈ F_{F} is good at level 2 if it satisfies the relation

(∗)  ∑_{i} n_{i} [x_{i}] ∧ [1 − x_{i}] = 0  in Λ^{2}(F^{×}) ⊗_{ℤ} ℚ ,

i.e. if the sum ∑ n_{i}[x_{i}] ∧ [1 − x_{i}] is a torsion element in the exterior square of F^{×}. (The element [x_{i}] ∧ [1 − x_{i}] is to be interpreted as 0 if x_{i} = 0, 1 or ∞.) Explicitly, (∗) means that if we pick a basis p_{1}, …, p_{s} of the subgroup of F^{×} generated by all x_{i} and 1 − x_{i} (modulo torsion) and write

x_{i} = ζ_{i} ∏_{j=1}^{s} p_{j}^{a_{ij}}, 1 − x_{i} = ζ′_{i} ∏_{j=1}^{s} p_{j}^{a′_{ij}}

where ζ_{i}, ζ′_{i} ∈ F^{×} are roots of unity and a_{ij}, a′_{ij} belong to ℤ, then

∑_{i} n_{i} (a_{ij} a′_{ik} − a_{ik} a′_{ij}) = 0 (1 ≤ j < k ≤ s).

In particular, if for some finite subset S = {p_{1}, …, p_{s}} of F^{×} the set of x ∈ F for which both x and 1 − x belong to

⟨S⟩ = { ζ p_{1}^{a_{1}} … p_{s}^{a_{s}} | ζ ∈ F^{×} a root of unity, a_{1}, …, a_{s} ∈ ℤ}

has cardinality bigger than \binom{s}{2}, then we can find good combinations of these elements x. A key question, both for finding relations among special values and for finding functional equations of polylogarithms, will be to find as large sets as possible of (numbers or rational functions) x ∈ F for which all x and 1 − x belong to a subgroup of F^{×} of small rank.

For any vector space, we can identify Λ^{2}(V) with V ⊗ V/Sym^{2}(V), where Sym^{2}(V) is the subspace of V ⊗ V spanned by elements x ⊗ y + y ⊗ x (or by elements x ⊗ x). Thus (∗) can be interpreted as saying that ∑ n_{i}[x_{i}] ⊗ [1 − x_{i}] belongs to Sym^{2}(F^{×}) up to torsion. The generalization to higher m is to say that a combination ξ ∈ F_{F} is good at level m if it satisfies the relation

(∗_{m})  ∑_{i} n_{i} \underbrace{[x_{i}] ⊗ … ⊗ [x_{i}]}_{m−1} ⊗[1 − x_{i}] ∈ Sym^{m}(F^{×}) ⊗_{ℤ} ℚ ,

where Sym^{m}(A) for any abelian group A is the 𝔖_{m}-invariant subspace of the mth tensor power A^{⊗m}. We can write the (m − 1)-fold tensor product [x] ⊗ … ⊗ [x] as [x]^{⊗(m−1)} or more simply, since it automatically belongs to the subspace Sym^{m−1}(F^{×}) of ⨂^{m−1}(F^{×}), as [x]^{m−1}. Thus the set G_{m}(F) of good combinations at level m is the kernel of the linear map

β_{m} : F_{F} → ((Sym^{m−1}(F^{×}) ⊗ F^{×})/Sym^{m}(F^{×})) ⊗_{ℤ} ℚ

defined on generators by

β_{m}([x]) = \{ \begin{matrix}[x]^{m−1} ⊗ [1 − x] (mod Sym^{m}(F^{×})) & if x ≠ 0, 1, ∞ , \\ 0 & if x = 0, 1, ∞ . \end{matrix}

We can also write G_{m}(F) in a way more analogous to the definition for m = 2 as the kernel of the map [x] ↦ [x]^{m−2} ⊗ ([x] ∧ [1 − x]) ∈ Sym^{m−1}(F^{×}) ⊗ Λ^{2}(F^{×}), since for any vector space V there is a natural injection Sym^{m−1}(V) ⊗ V/Sym^{m}(V) → Sym^{m−2}(V) ⊗ Λ^{2}(V).

The first relation between G_{m}(F) and the polylogarithm function is that functional equations of Li_{m} modulo lower order polylogarithms are given by combinations ξ = ∑ n_{i}[x_{i}] of rational functions x_{i} = x_{i}(t) satisfying (∗_{m}). Since Li_{m} cannot be extended to a continuous one-valued function on the whole complex plane, we will work instead with the modified function

P_{m}(x) = ℜ_{m}( ∑_{j=0}^{m−1} \frac{2^{j} B_{j}}{j!} (log |x|)^{j} Li_{m−j}(x));

here ℜ_{m} denotes ℑ or ℜ according as m is even or odd and B_{j} the jth Bernoulli number (B_{0} = 1, B_{1} = −\frac{1}{2}, B_{2} = \frac{1}{6}, B_{3} = 0, …). This function is continuous and real-valued on ℙ^{1}(ℂ) = ℂ ∪ {∞} and is real-analytic except at 0, 1 and ∞. (The function P_{2}(x) is just the Bloch-Wigner function D(x) defined in 1.5.1 of this book.) Using P_{m} instead of Li_{m} also has the advantage that the lower order terms in all functional equations cancel, so that ∑_{i} P_{m}(x_{i}(t)) =constant for any good combination ∑ n_{i}[x_{i}(t)]. The proof, which is not difficult, is given in §7 of [Z] and will not be repeated here. As simple examples, we note that

β_{m}(\frac{1}{x}) = (−[x])^{m−1} ⊗ ([−1] + [1 − x] − [x]) = (−1)^{m−1}β_{m}(x)

for any x ∈ F, since [−1] is a torsion element of F^{×} and [x]^{m−1}⊗[x] belongs to Sym^{m}(F^{×}), and similarly

β_{m}(x^{2}) = (2 [x])^{m−1} ⊗ ([1 − x] + [1 + x]) = 2^{m−1} (β_{m}(x) + β_{m}(−x))

or more generally β_{m}(x^{N}) = ∑_{ζ^{N} =1} β_{m}(ζx) for any N ≥ 1, corresponding to the inversion and distribution relations

P_{m}(1/x) = (−1)^{m−1}P_{m}(x), P_{m}(x^{N}) = N^{m−1} ∑_{ζ^{N} =1} P_{m}(ζx)

of the polylogarithm. Many other examples will be discussed in §7.

We now return to number fields and the definition of A_{m}(F). If ϕ is any homomorphism from F^{×} to ℤ (e.g. ϕ = v_{p} for F = ℚ, where p is a prime number and v_{p}(x) denotes the power of p contained in a rational number x), then for m > 2 the map ι_{ϕ} : F_{F} → F_{F} sending [x] to ϕ(x) [x] sends G_{m}(F) to G_{m−1}(F) (because β_{m−1} ∘ ι_{ϕ} = (ϕ ⊗ id^{⊗(m−1)}) ∘ β_{m} under the natural identification of ℤ ⊗ (F^{×})^{⊗(m−1)} with (F^{×})^{⊗(m−1)}). If we have r homomorphisms ϕ_{i} (1 ≤ i ≤ r), then for m ≥ r + 2 the composite map ι_{ϕ_{1}} ∘ … ∘ ι_{ϕ_{r}} : G_{m}(F) → G_{m−r}(F) is independent of the order of the ϕ_{i} and will be denoted simply ι_{ϕ_{1}} … ι_{ϕ_{r}}. The elements ι_{ϕ_{1}} … ι_{ϕ_{r}}(ξ) ∈ G_{m−r}(F) (1 ≤ r ≤ m − 2) for an element ξ of G_{m}(F) will be called the good combinations associated to ξ. Up to linear combinations there are only finitely many of them, since only the restrictions of the ϕ_{i} to the subgroup of F^{×} generated by the elements occurring in ξ are important and we can let ϕ_{i} run over a basis of the group of homomorphisms from this group to ℤ.

The definition of A_{m}(F) is now as follows. We start with A_{2}(F) = G_{2}(F). For ξ = ∑ n_{i}[x_{i}] ∈ A_{2}(F) and each embedding σ : F ↪ ℂ, we consider P_{2}(ξ^{σ}), where ξ^{σ} denotes ∑ n_{i}[x_{i}^{σ}] and P_{2}(ξ^{σ}) denotes ∑ n_{i}P_{2}(x_{i}^{σ}) (we shall use this abbreviated notation for the value of functions on combinations of arguments from now on). There are only r_{2} essentially distinct such values, since the relation P_{2}(\overline{x}) = −P_{2}(x) for x ∈ ℂ implies that P_{2}(ξ) vanishes for the real embeddings of F and is the same up to sign for each conjugate pair of complex embeddings. Thus, picking one of each conjugate pair of non-real embeddings, we get a map P⃗_{2} from A_{2}(F) to ℝ^{r_{2}} by sending ξ to {P_{2}(ξ^{σ})}_{σ}. The image P⃗_{2}(A_{2}(F)) is contained in a lattice (= discrete subgroup of maximal rank) R_{2} = R_{2}(F) ⊂ ℝ^{r_{2}}. Therefore, if A_{2}(F) has rank > r_{2} (and we will see later that its rank is actually infinite), then the kernel C_{2}(F) of P⃗_{2} contains non-zero elements and we get non-trivial relations among dilogarithms of complex algebraic arguments. Furthermore, if ξ ∈ C_{2}(F) and σ : F ↪ ℝ is a real embedding of F, then L_{2}(ξ^{σ}), where L_{2}(x) = Li_{2}(x)+ \frac{1}{2} log |x| log |1−x| (x ∈ ℝ) is the Rogers dilogarithm function (cf. 1.2.3), is a rational multiple of π^{2}, so we also get relations among dilogarithms of real algebraic arguments. For m = 3 we define A_{3}(F) as the subset of G_{3}(F) consisting of all good combinations ξ for which ι_{ϕ}(ξ) ∈ A_{2}(F) belongs to C_{2}(F) for every homomorphism ϕ : F^{×} → ℤ. This condition can be checked by numerical computation since the element P_{2}(ι_{ϕ}(ξ)) belongs to the discrete group R_{2} and hence is recognizably zero or non-zero. For ξ ∈ A_{3}(F) we consider the collection of all P_{3}(ξ^{σ}) where (since P_{3}(x) is invariant rather than anti-invariant under complex conjugation of x) σ now ranges over the real and half the complex embeddings of F. This defines a map P⃗_{3} : A_{3}(F) → ℝ^{r_{1}+r_{2}} whose image is again contained in a lattice R_{3} ⊂ ℝ^{r_{1}+r_{2}}, so again we get a non-trivial kernel C_{3}(F) and non-trivial relations among polylogarithm values. Similarly, for m > 2 we consider the map P⃗_{m} : F_{F} → ℝ^{n_{∓}}, where ±1 = (−1)^{m}, n_{+} and n_{−} denote r_{1} + r_{2} and r_{2} respectively, and P⃗_{m}(ξ) is the collection of P_{m}(ξ^{σ}) with σ ranging over half the complex embeddings (m even) or all the real embeddings and half the complex embeddings (m odd). The subgroup A_{m}(F) ⊆ G_{m}(F) ⊂ F_{F} will be defined inductively in such a way that its image under P⃗_{m} is contained in a certain lattice R_{m} = R_{m}(F) ⊂ ℝ^{n_{∓}}, in which case the kernel

C_{m}(F) = {ξ ∈ A_{m}(F) | P_{m}(ξ^{σ}) = 0 for all σ : F ↪ ℂ }

is a subgroup of corank at most n_{∓}. Assuming that A_{m′}(F) and C_{m′}(F) have been defined for m′ < m, we define A_{m}(F) by

A_{m}(F) = {ξ ∈ G_{m}(F) | ι_{ϕ}(ξ) ∈ C_{m−1}(F) for all homomorphisms ϕ : F^{×} → ℤ } .

Explicitly, this means that starting with ξ ∈ G_{m}(F), we verify ξ ∈ A_{m}(F) by checking first that all associated elements ι_{ϕ_{1}} … ι_{ϕ_{m−2}}(ξ) ∈ A_{2}(F) map to 0 under P⃗_{2}, then that all associated elements ι_{ϕ_{1}} … ι_{ϕ_{m−3}}(ξ) (which are then known to belong to A_{3}) map to 0 under P⃗_{3}, and so on in succession for all associated elements ι_{ϕ_{1}} … ι_{ϕ_{r}}(ξ) for r = m − 2, m − 3, …, 2, 1.

We do not give any further explanation of these definitions here; the examples in the following sections should make it clear how they work in practice, while the background and motivating ideas are explained in more detail in [Z]. However, we should say a few words about the relationship to K-theory and about what has been proved so far and by whom. The former can be understood without needing to know the definition of algebraic K-groups K_{n}(F). One need only know that there are such groups and that by a result of Borel [B], the even-index groups are finite while the odd-index groups K_{2m−1}(F) (m > 1) are mapped isomorphically (up to torsion) onto a lattice in ℝ^{n_{∓}} by the so-called “regulator mapping.” The lattice R_{m}(F) we have been speaking about is the image of the regulator mapping (or more precisely, any lattice commensurable with this image, since everything we are saying is known only up to groups of finite order). Borel also showed that the covolume of R_{m} ⊂ ℝ^{n_{∓}} is a rational multiple of |D_{F} |^{1/2} ζ_{F} (m)/π^{mn_{±}}. Hence if the map P⃗_{m} : A_{m}(F) → R_{m} is surjective, then ζ_{F} (m) can be expressed in terms of polylogarithms of order m with arguments in F.

The relationship between the dilogarithm and the regulator lattice R_{2} was found by Bloch (for this and later developments concerning K_{3} and dilogarithms, see Suslin’s ICM talk [S]). In this case the map P⃗_{2} : A_{2} → R_{2} is surjective (after tensoring with ℚ) and its kernel is the subgroup of A_{2}(F) spanned by the 5-term functional equation of the dilogarithm. The above picture for general m was formulated conjecturally in [Z]. The case m = 3 was proved completely by Goncharov [G], who showed that the map P⃗_{3} : A_{3}(F) → ℝ^{n_{+}} maps not only into, but onto the regulator lattice R_{3} (at least after tensoring with ℚ) and also gave a complete description of the kernel in terms of a new 22-term functional equation for the trilogarithm. The fact that P⃗_{m} maps A_{m}(F) into the Borel regulator lattice for arbitrary m was proved by Deligne and by Beilinson for cyclotomic fields and by Beilinson in the general case [BD]. For m > 3 the surjectivity is not known except in the cyclotomic case, so that the desired corollary that ζ_{F} (m) can be expressed in terms of special values of polylogarithms at arguments in F is still a conjecture in general. We also conjecture that C_{m}(F) can be described completely in terms of the functional equations of the mth polylogarithm, but this also is known only for m = 2 and 3.

Finally, we should mention that the generalized Rogers function L_{m}(x) (x ∈ ℝ) defined in 1.5.1 agrees with P_{m} for m = 3 and x real, while for m > 3 odd and x real it is equal to ∑_{r=0}^{m−2}(log |x|)^{r} P_{m−r}(x)/(r + 1)! (only the terms with r even in this sum contribute, since P_{m′}(x) = 0 for m′ even and x real). Thus for elements ξ ∈ A_{m}(F) (m odd) and real embeddings σ of F, the elements P_{m}(ξ^{σ}) and L_{m}(ξ^{σ}) agree (because each term ∑ n_{i}(log |x_{i}|)^{r}P_{m−r}(x_{i}), r > 0, vanishes by virtue of the inductive definition of A_{m}), and therefore our relations among values of P_{m}(x) at real algebraic arguments can be reinterpreted as relations among the same values of L_{m}(x). For m even, P_{m}(x) vanishes identically for x real, but if we take an element ξ ∈ A_{m}(F) for which P_{m}(ξ^{σ}) = 0 for all complex embeddings σ, then it is apparently a consequence of the results in [BD] that each L_{m}(ξ^{σ}) (σ : F ↪ ℝ) is a rational multiple of π^{m}.

2. Examples of dilogarithm relations. For m = 2 there is no distinction between G_{m} and A_{m}, so we just have to look for good combinations at level 2, i.e., combinations ξ = ∑ n_{i}[x_{i}] satisfying (∗). For such ξ, P_{2}(ξ) = D(ξ) belongs to the r_{2}-dimensional lattice R_{2}. Thus if we find r_{2} elements ξ with linearly independent images, we obtain ζ_{F} (2) as π^{2(r_{1}+r_{2})}/\sqrt{|D_{F} |} times a rational number times an r_{2} × r_{2} determinant of integral linear combinations of dilogarithm values, and if we have more than r_{2} good combinations ξ, then they will have linearly dependent images in the lattice and we obtain linear relations over ℚ among the values D(x_{i}) (resp. L_{2}(x_{i}^{σ}) modulo π^{2} for the real embeddings σ of F, where L_{2}(x) is the Rogers dilogarithm).

As an example over ℚ, take elements x ∈ ℚ such that x and 1 − x contain no primes except 2, 3, 5 and 7. There are exactly 375 such x, forming 63 orbits under the group generated by x ↦ 1/x, x ↦ 1−x. For each one, β_{2}(x) = [x]∧[1−x] is a linear combination of the six elements

[2] ∧ [3], [2] ∧ [5], [2] ∧ [7], [3] ∧ [5], [3] ∧ [7], [5] ∧ [7],

so we get 57 essentially different linearly independent linear combinations ξ belonging to A_{2}(ℚ), for each of which L_{2}(ξ) is a rational multiple of π^{2}. For instance, if we pick at

random the seven elements \frac{1}{3}, −\frac{1}{6}, \frac{2}{7}, \frac{1}{8}, \frac{1}{9}, \frac{1}{21}, and \frac{1}{28} (any other seven would do just as well), then since 7 > 6 we must find at least one non-trivial element of A_{2}(ℚ). In fact, calculating

β_{2}(\frac{1}{3}) = [\frac{1}{3}] ∧ [\frac{2}{3}] = [2] ∧ [3], β_{2}(−\frac{1}{6}) = [−\frac{1}{6}] ∧ [\frac{7}{6}] = −[2] ∧ [7] − [3] ∧ [7], etc.,

we find that two linearly independent combinations of the 7 elements in question are annihilated by β_{2}, namely 6[\frac{1}{3}] − [\frac{1}{9}] and 3[−\frac{1}{6}] − [\frac{1}{8}] + [\frac{1}{9}] + [\frac{1}{28}]. The image of each of these under the Rogers dilogarithm must be a rational multiple of π^{2}, and indeed,

6 L_{2}(\frac{1}{3}) − L_{2}(\frac{1}{9}) = \frac{π^{2}}{3} , 3 L_{2}(−\frac{1}{6}) − L_{2}(\frac{1}{8}) + L_{2}(\frac{1}{9}) + L_{2}(\frac{1}{28}) = −\frac{π^{2}}{12} .

Taking instead a field with r_{2} > 0, we let F = ℚ(\sqrt{−7}). Then for the two elements x_{1} = (1 + \sqrt{−7})/2 and x_{2} = (−1 + \sqrt{−7})/4, both 1 − x_{1} and 1 − x_{2} belong to the group generated by −1, x_{1}, and x_{2}, so both β(x_{1}) and β(x_{2}) are multiples of [x_{1}] ∧ [x_{2}]. The multiples turn out to be 1 and −2, respectively, so 2[x_{1}] + [x_{2}] belongs to A_{2}(F). Hence 2D(x_{1}) + D(x_{2}) must be a rational multiple of ζ_{F} (2)/π^{2}\sqrt{7}, and indeed one finds

2D(\frac{1 + \sqrt{−7}}{2}) + D(\frac{−1 + \sqrt{−7}}{4}) = \frac{21\sqrt{7}}{4π^{2}} ζ_{ℚ(\sqrt{−7})}(2).

As a third example, we consider the number u = \sqrt{ρ}, where ρ = (\sqrt{5}−1)/2 (cf. Chapter 5, section 5.2.1). Here the relation (5.4) implies that ξ_{0} = [u^{6}] − 4[u^{3}] + 6[u] is a good element of F_{F}, F = ℚ(u), since

β_{2}(ξ_{0}) = [u^{6}] ∧ [1 − u^{6}] − 4[u^{3}] ∧ [1 − u^{3}] + 6[u] ∧ [1 − u]
= 6[u] ∧ [(1 − u^{6})(1 − u^{3})^{−2}(1 − u)] = 12[u] ∧ [u] = 0.

However, ξ_{0} does not belong to the subgroup C_{2}(F) of A_{2}(F), because, denoting by σ the embedding of F into ℂ which sends u to u^{σ} = i/u, we have D(ξ_{0}^{σ}) = 8.6124152 … ≠ 0. Therefore we should not expect L_{2}(ξ_{0})/π^{2}, which is the number given by (5.6), to be rational. What we do expect is that D(ξ_{0}^{σ}) is a rational multiple of π^{−6}ζ_{F} (2), since for the field F we have r_{1} = 2, r_{2} = 1, D_{F} = 400. To compute numerically, we decompose ζ_{F} (s) as ζ_{F_{1}}(s)L_{F_{1}}(s) where ζ_{F_{1}}(s) is the Dedekind zeta function of F_{1} = ℚ(\sqrt{5}) and

L_{F_{1}}(s) = \frac{1}{1 + 5^{−s}} ∏_{p≡11, 13, 17, 19} \frac{1}{1 − p^{−2s}} ∏_{p≡3, 7} \frac{1}{1 + p^{−2s}} ∏_{p≡1, 9}(\frac{1}{1 + ε_{p}p^{−s}} )^{2}

(here the congruences on primes p are modulo 20 and ε_{p} = (\frac{ρ}{p}) = (\frac{2i + 1}{p}) for p ≡ 1

or 9 mod 20, where ρ and i are solutions of ρ^{2} + ρ − 1 ≡ i^{2} + i ≡ 0 (mod p)). We have ζ_{F_{1}}(2) = 20π^{4}/75\sqrt{5} and (computing numerically with the Euler product, using all primes up to 2500) 20^{3/2}π^{−2}L_{F_{1}}(2) ≈ 8.612406, agreeing with D(ξ_{0}^{σ}) to the accuracy of the computation. If we want to find elements of C_{2}(F) and hence relations among the Rogers dilogarithms of elements of F, then we must look at combinations involving other numbers of F than just powers of u, i.e., it is not enough to consider ladders only. For instance, the group of units of F is generated (up to torsion) by u and v = 1 − u, and we can look at elements x ∈ F for which both x and 1 − x belong to this group. Up to equivalence by the group generated by x ↦ 1/x and x ↦ 1−x, under which D(x) is invariant up to sign, there are 5 such elements, namely x = u, −u, u^{2}, u^{2}v and v^{2}/u^{3}. For these elements 1−x equals v, u^{4}/v, u^{4}, u^{7}/v and v/u^{6}, respectively, so β_{2}[x] = [x]∧[1−x] equals [u]∧[v] times 1, −1, 0, −9 and 9, respectively. We therefore get four linearly independent elements [u]+[−u], [u^{2}], [u^{2}v] + 9[u] and [u^{2}v] + [v^{2}/u^{3}] belonging to A_{2}(F). The first two of these are proportional by the duplication formula and uninteresting because they reduce to relations from the smaller field F_{1}. Computing numerically, we find that D((u^{2}v)^{σ}) + 9D(u^{σ}) equals D(ξ_{0}^{σ}) but that D((u^{2}v)^{σ}) + D((v^{2}/u^{3})^{σ}) vanishes. Hence [u^{2}v] + [v^{2}/u^{3}] and [u^{2}v] + 9[u] − ξ_{0} belong to C_{2}(F) and should map under the Rogers dilogarithm to rational multiples of π^{2}, and indeed,we have

L_{2}(u^{2}v) + L_{2}(v^{2}/u^{3}) = \frac{1}{20} π^{2} , L_{2}(u^{2}v) + 9L_{2}(u) − L_{2}(ξ_{0}) = \frac{41}{60} π^{2} .

3. Examples for higher order polylogarithms. Many examples of relations among trilogarithms of algebraic arguments are given in §§3–5 of [Z] as motivation for the form of the conjectures explained in §1. Here we give instead two examples for the field F = ℚ (also both taken from [Z]). The first illustrates the necessity of the extra condition in the definition of A_{m} as opposed to G_{m}, while the second gives an example of a relation for heptalogarithms, beyond the range of known functional equations.

For the first example, we proceed as in the example for m = 2 in §3, but using only the primes 2 and 3 (if we used 2, 3, 5 and 7 again, we would get many more examples). There are 21 numbers x ∈ ℚ for which x and 1 − x contain only the prime factors 2 and 3, but this number is cut down to 11 if we do not take both x and x^{−1} (for which the values of all polylogarithms are the same up to sign), and further cut down to 8 if we eliminate the numbers −1, 1/4 and 1/9 for which the polylogarithm reduces to simpler values by virtue of the duplication equation. These values are 1/2, 1/3, 2/3, −1/2, 3/4, −1/3, 8/9 and −1/8. The image of each of them under β_{5} belongs to a space of dimension 4 (this is the dimension of Sym^{4}(V)⊗V/Sym^{5}(V) for V the 2-dimensional subspace of ℚ^{×} spanned by 2 and 3), so we must have at least four independent elements ξ ∈ Ker(β_{5}), and indeed there are exactly four, namely [1/2], [−1/3] − 2[1/3], [−1/8] − 162[−1/2] and [8/9] − 9[3/4] − 36[2/3]−18[−1/2]−6[1/3]. This gives a 4-dimensional subspace of G_{5}(ℚ). However, to get elements of A_{5}(ℚ), we need three further conditions, namely that the images of ξ in A_{3}(ℚ) under the three maps sending [x] to v_{2}(x)^{2}[x], v_{2}(x)v_{3}(x)[x], and v_{3}(x)^{2}[x] all map to zero under the trilogarithm map P⃗_{3} : A_{3}(ℚ) → ℚζ(3). This cuts down the dimension from four to one, the unique surviving relation being ξ = [−1/8] − 126[1/2] − 162[−1/2]. This element should therefore map to a rational multiple of ζ(5) under P_{5} or L_{5}, and indeed we find that P_{5}(ξ) = L_{5}(ξ) = \frac{403}{16} ζ(5) .

To get an example for the heptalogarithm takes more work. If we consider the set of all x such that both x and 1 − x contain only the first s primes for some s (as we did for m = 2 with s = 4 and for m = 5 with s = 2), then the number of conditions we have to satisfy is so large that the first value of s for which there are enough x to give a non-trivial element of A_{7}(ℚ) is 8 (i.e., use all primes less than 20), for which there are 10946 elements x (up to inversion) and “only” 10662 conditions to be satisfied (cf. [Z], §10A). This system of equations is far too large to solve numerically. Instead we consider x ∈ ℚ for which x ∈ ⟨{2, 3}⟩ and 1 − x ∈ ⟨{2, 3, 5, 7}⟩. There are 29 such x (taking only one of each pair x, 1/x and omitting squares as before). On the other hand, there are 28 conditions to be fulfilled: first 20 to get good combinations at level 7 (if V and W are the 2- and 4-dimensional subspaces of ℚ^{×} ⊗_{ℤ} ℚ generated by {2, 3} and {2, 3, 5, 7}, respectively, the

number of conditions is dim(Sym^{6}(V)⊗W/Sym^{7}(V)) = \binom{7}{6}·4−\binom{8}{7}) and then a further 8 to ensure that each of the combinations ι_{v_{2}}^{j}ι_{v_{3}}^{r−j}(ξ) (r = 4 or 2, 0 ≤ j ≤ r) maps to 0 in the one-dimensional lattice R_{7−r}. Since 29 is bigger than 28, we must find a solution of this system of equations, i.e., a combination ξ ∈ A_{7}(ℚ). This solution turns out to be unique up to multiplication by a constant and is given by ξ = ∑ n_{i}[x_{i}] ∈ A_{7}(ℚ), where the n_{i} (normalized for convenience to be in \frac{1}{6}ℤ rather than ℤ) and x_{i} are given by the following table:

n_{i}  x_{i}  n_{i}  x_{i}  n_{i}  x_{i}
−25111753072  1/3  −284585110 1/8  1765911 3/128
−27461584367 −1/3  470985412/3 −1/8  478706760 2/9
−171330250 −1/9  38987641 −1/24  −66158750 8/9
57577037 −1/27  17015061/2 −1/48  15912813 −9/16
−151540388696  1/2  −11528187258 2/3  23786119 2/27
−136446322032 −1/2  −6563312469 −2/3  −2879429 −8/27
−2209899405  1/6  2802854628 3/4  2585366 27/32
−2199243270 −1/6  −751304106 −3/4  111363 32/81
43524 −1/4374  −785318380 3/8  −2372265 −2/243
−7089743800 −1/4  11883921 −3/32

The theory now predicts that ∑ n_{i}P_{7}(x_{i}) should be a rational multiple of ζ(7), and indeed,

computing numerically we find that it equals −\frac{1020149599795}{96} ζ(7) to high precision.

4. Examples: ladders. We now come to the subject of Leonard Lewin’s ladders (note once again the fascination with the letter “L” which marks this field; cf. [L], p. 191), the source of most of the examples in the book, and show how they fit into the theory sketched so far. Briefly, ladders are the special case of combinations ξ = ∑ n_{i}[x_{i}] in which all of the x_{i} are powers of a single number α. The advantage of making this restriction is that the conditions needed to make ξ belong to A_{m} become much more transparent and easier to check (and to fulfill) than in the general case. The disadvantage, of course, is that it is very difficult and requires great ingenuity to produce examples, whereas providing examples of general elements in G_{m} or A_{m} is something which can be done in a mechanical manner. In particular, while we can show (§5) that there exist polylogarithmic relations for arbitrarily high orders m, it is not at all clear—indeed, rather unlikely—that there are any valid ladders at all with m larger than, say, 20.

Suppose, then, that all of the x_{i} are powers of a single number α. Since [x] and [1/x] are essentially equivalent for all polylogarithmic purposes, we can restrict to positive powers. Then the combinations ξ we are looking for can be written as

(1)  ξ = ∑_{j=1}^{∞} n_{j} [α^{j}]  (n_{j} ∈ ℤ, n_{j} = 0 for all but finitely many j) .

(From now on we will abbreviate the condition in brackets as {n_{j}} ∈ ⊕_{j≥1} ℤ.) If we compute

the image of this under the map β_{m}, we find (since [α^{j}] = j[α] in F^{×})

β_{m}(ξ) = ∑_{j≥1} j^{m−1} n_{j} [α]^{m−1} ⊗ [1 − α^{j}] = [α]^{m−1} ⊗ [∏_{j≥1}(1 − α^{j})^{j^{m−1}n_{j}}].

This is 0 modulo Sym^{m}(⟨α⟩) if and only if ∏_{j}(1 − α^{j})^{j^{m−1}n_{j}} is (a root of unity times) a power of α. In other words, turning things around, given any cyclotomic relation

(2)  ∏_{j=1}^{∞}(1 − α^{j})^{b_{j}} = (root of 1) × α^{N}  ({b_{j}} ∈ ⊕_{j≥1} ℤ, N ∈ ℤ)

we get good combinations ∑_{j=1}^{∞} j^{−m+1} b_{j} [α^{j}] for all levels m. (Of course, if we want integral

combinations, we must multiply these by J^{m−1}, where J is a common multiple of the j with n_{j} ≠ 0.) This is the beauty of ladders: they give an immediate construction of elements of G_{m}(F) for all m simultaneously. However, to get relations among polylogarithm values, we need combinations in A_{m}, not G_{m}. Here, too, the special property that V = ⟨{x_{i}}⟩ is 1-dimensional simplifies life considerably, since it means that there is up to scalar factors only one homomorphism ϕ from V to ℚ (namely, the one sending α^{j} to j) and hence only one way to associate to a good combination ξ = ∑ n_{j}[α^{j}] of order m good combinations of smaller order m′ = m − r, namely ξ′ = ι_{ϕ}^{r}(ξ) = ∑_{j} j^{r}n_{j}[α^{j}]. Thus we get the following

inductive picture. Let {b_{j}^{(1)}}, …, {b_{j}^{(d)}} be a multiplicatively independent set of cyclotomic relations (2) for the same number α. Then for each m we get d linearly independent elements ξ_{m}^{(ν)} = ∑ j^{−m} b_{j}^{(ν)} [α^{j}] of G_{m}(F), where F = ℚ(α). For m = 2 each of these elements belongs to A_{2} and maps to the r_{2}-dimensional lattice R_{2}, so we get at least d−r_{2} dilogarithm relations. Changing our basis for the set of cyclotomic relations we can assume that these are ξ_{2}^{(1)}, …, ξ_{2}^{(d−r_{2})}, i.e., for each ν ≤ d − r_{2} we have ∑ j^{−2}b_{j}^{(ν)}D((α^{σ})^{j}) = 0 if α^{σ} is complex, ∑ j^{−2}b_{j}^{(ν)}L_{2}((α^{σ})^{j}) ∈ ℚ π^{2} if α^{σ} is real, and ξ_{3}^{(ν)} = ∑ j^{−3}b_{j}^{(ν)}[α^{j}] ∈ A_{3}(F). Now the elements ξ_{3}^{(1)}, …, ξ_{3}^{(d−r_{2})} map under D_{3} to the (r_{1} + r_{2})-dimensional lattice R_{3}, so we get at least d − r_{2} − (r_{1} + r_{2}) = d − n (n = [F : ℚ]) linearly independent relations among the trilogarithms of the α^{j} and the same number of elements in the next higher group A_{4}(F). Continuing in this way, we find that the dimension goes down alternately

by r_{2} and r_{1} +r_{2}, so by n every two steps. Hence after approximately \frac{2d}{n} steps the process terminates, unless we are very lucky (i.e., there happen to be more linear relations than is forced by the number of equations to be satisfied, a rather rare occurrence) or d is infinite. In fact the second alternative cannot happen, since one can show that the number

d(α) =_{DEF} rk_{ℤ}{{b_{j}} ∈ ⊕_{j≥1} ℤ | ∏_{j=1}^{∞}(1 − α^{j})^{b_{j}} ∈ ⟨α⟩}

is finite for any algebraic number α which is not a root of unity (of course d(α) = 0 if α is transcendental).

We now illustrate all of this with two examples. We first take α to be the number ω treated in [AL] and in Chapter 4 of the present book, i.e.,the root of ω^{3} = ω +1. Here d(α) is (at least) 12, the corresponding special exponents (= exponents j for which a positive power of 1 − α^{j} belongs to the subgroup generated by α and 1 − α^{i} with i < j) being 1, 2, 3, 5, 8, 12, 14, 18, 20, 28, 30 and 42. Since r_{1} = r_{2} = 1, we can go up to m = 8, the number of linear relations obtained for the mth order polylogarithms being 11, 9, 8, 6, 5, 3, 2 for m = 2, 3, …, 8, respectively. At the 9th level there are no more relations. However, if we include 1 as well as the positive powers of ω then there is one more relation in odd levels and in particular we get an expression of ζ(9) as a rational linear combination of D_{9}(ω^{n}) with n > 0. More details of this example can be found in Chapter 4 and in §9C of [Z].

The most spectacular example is the Salem number treated in Chapter 16 (section 16.3), namely the solution of α^{10} + α^{9} − α^{7} − α^{6} − α^{5} − α^{4} − α^{3} + α + 1 = 0. Here d(α) is (at least, and probably exactly) 71, the special exponents ranging up to j = 360. Here r_{2} = 4 and r_{1} + r_{2} = 6, so the number of conditions to be satisfied at the even and odd steps is a priori 4 and 6, respectively, but because the two real conjugates of α are inverses of one another and D_{m}(x^{−1}) = D_{m}(x) for m odd, there are in fact only 5 independent conditions at odd levels. We therefore get successively (at least) 71, 67, 62, 58, … elements of R_{2}, R_{3}, R_{4}, R_{5}, …. Thus the ladder reaches up to m = 16.

By the way, to find the special exponents, one proceeds as follows. First calculate the norm of α^{j} − 1 by multiplying the conjugates numerically for j up to, say, 1000. The norm is 1 for the 22 values j = 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 21, 23, 27, 29, 34, 37, 47, 63, 65 and 74. Since the unit rank is r_{1} + r_{2} − 1 = 5, we lose 4 relations, because α, 1 − α, 1 − α^{2}, 1 − α^{3} and 1 − α^{5} are needed to generate a group of units of full rank, but we then get 18 independent multiplicative relations among α and the 1 − α^{j}. Now we look at other j for which the norm of 1−α^{j} is no longer 1 but still factors into small prime factors which have already occurred for previous j, and then try to form combinations which are units. For instance, the norms of α^{j} − 1 are positive powers of 3 for j = 4, 8, 12, 16, 20, 24, 36, and 40, and eliminating α^{4} − 1, which together with the units generates the group of all elements of F whose norm is a power of 3, we get 7 further relations. Proceeding in this way (in practice it is more efficient to use the cyclotomic polynomials Φ_{j}(α) rather than the numbers α^{j} − 1) gives the 71 relations mentioned. By the time we get to j = 1000, the norms are so huge that it seems clear that α^{j} − 1 will never again be a combination of smaller values, and this could be proved by a finite effort if really required.

It seems very likely that this particular α gives the maximum of \frac{d(α)}{[ℚ(α) : ℚ]} (indeed,

quite possibly even of d(α)) for all algebraic numbers, in which case there are probably no valid ladders at all of order bigger than 16. In any case, the study of the number d(α), motivated by the ladder concept, seems to be a very interesting problem in the field of diophantine approximation.

5. Existence of relations among polylogarithm values of arbitrarily high order. Let S be a set of s numbers in ℚ and X(S) the set of all x ∈ ℚ such that both x and 1 − x belong to the multiplicative group generated by the elements of S. The number of independent requirements on a combination ξ = ∑_{x∈X} n_{x}[x] to belong to A_{m}(ℚ) is

(m − 1)\binom{s + m − 2}{m} + ∑_{\begin{matrix}0≤r<m−2 \\ m−r odd \end{matrix}} \binom{s + r − 1}{r}

(the first term is the dimension of Sym^{m−1}(V) ⊗ V/Sym^{m}(V) where V = ⟨S⟩ is sdimensional; the other terms give the number of successive conditions for the images of ξ under the various maps ι_{ϕ_{i_{1}}} … ι_{ϕ_{ir}} to map to 0 in R_{m−r} ≅ ℤ), which is a polynomial in s. Therefore if there are sets S of arbitrarily large cardinality s for which |X(S)| grows more than polynomially with s, then it follows that A_{m}(ℚ) has infinite rank and hence that there are infinitely many relations among the values of L_{m}(x) (x ∈ ℚ) for every m. The existence of such S is the content of a theorem of Erdös-Stewart-Tijdeman [EST]. A simplified presentation of their proof was given in [Z] and a further simplification (with a slightly weaker bound, but not appealing to any results from analytic number theory) is given here.

Theorem (Erdös-Stewart-Tijdeman). For any ε > 0 there exist sets S of arbitrarily large cardinality s for which |X(S)| > e^{(2−ε)\sqrt{s/ log s}} .

Proof. Let t and u be large numbers, to be chosen presently, and let A denote the set of products p_{1} … p_{r} with r ≤ u and each p_{i} a prime ≤ t. The cardinality of A is then

\binom{u + π(t)}{u} which for π(t) (number of primes ≤ t) much larger than u equals \frac{π(t)^{u+o(u)}}{u!} =

(\frac{eπ(t)}{u})^{u+o(u)}. The number of pairs (a, c) with a, c ∈ A, c > a, is therefore equal to

\binom{|A|}{2} = (\frac{eπ(t)}{u})^{2u+o(u)}. For each such pair the difference b = c − a is an integer between

0 and t^{u}, so by the pigeonhole principle there is a number b which is expressible as c−a in at

least N = \binom{|A|}{2}/t^{u} = (\frac{eπ(t)}{u\sqrt{t}})^{2u+o(u)} ways. We maximize this by choosing u = π(t)/\sqrt{t},

giving N = e^{(2+o(1))π(t)/\sqrt{t}}. Now take S = \{b\} ∪ \{p ≤ t, p prime\}, with cardinality

s = 1 + π(t) = (1 + o(1))\frac{t}{log t}; then each representation b = c − a gives a distinct element

x = \frac{b}{c} of X(S), so |X(S)| > N = e^{(2+o(1))\sqrt{s/ log s}} .

Corollary. (1) A_{m}(ℚ) has infinite rank for all m. (2) For any m, there are infinitely many linearly independent relations over ℚ among the values of L_{m}(x), x ∈ ℚ.

Remarks. 1. The original result of Erdös-Stewart-Tijdeman was slightly stronger in that 2 was replaced by 4 in the exponent and S was required to consist only of primes; for the proof of the stronger statement, see [EST] or [Z].

2. In part (1) of the corollary, we could have written “A_{m}(F) for any number field F ” instead of A_{m}(ℚ), since A_{m}(F) contains A_{m}(ℚ), but of course the interesting question is whether one gets infinitely many new relations on passing from ℚ to F, and this is not clear without doing more work.

3. Of course, part (2) of the corollary is not a corollary of the Erdös-Stewart-Tijdeman theorem alone, but of this theorem together with Beilinson’s deep result on the relation between the polylogarithm and regulator mappings.

6. A conjecture on linear independence. We conjecture that the only linear relations over ℚ among polylogarithm values at algebraic arguments are those which follow from the theory explained in §1. More precisely, this says that if there is a relation

(1)  ∑_{i} n_{i}P_{m}(x_{i}) = 0  (n_{i} ∈ ℤ, x_{i} ∈ \overline{ℚ}) ,

(respectively ∑ n_{i}L_{m}(x_{i}) = 0 with n_{i} ∈ ℤ and x_{i} ∈ \overline{ℚ} ∩ ℝ), then (i) ∑ n_{i}x_{i} satisfies the algebraic relation (∗_{m}), and (ii) the conjugate equations ∑ n_{i}P_{m}(x_{i}^{σ}) = 0 (σ ∈Gal(\overline{ℚ}/ℚ)), as well as the associated equations ∑ n_{i}ϕ_{1}(x_{i}) ⋯ ϕ_{r}(x_{i})P_{m−r}(x_{i}) = 0 and their conjugates for all r ≤ m − 2 and all homomorphisms ϕ_{1}, …, ϕ_{r} : \overline{ℚ}^{×} → ℚ, also hold.

Since we further conjecture that the kernel of the maps P⃗_{m} : A_{m}(F) → ℝ^{n_{∓}} is the group C_{m}(F) defined by specializing functional equations, we can state the combined conjecture more concisely by saying that the only relations (1) are specializations of functional equations ∑ n_{i}P_{m}(ϕ_{i}(t)) = 0 (ϕ(t) ∈ ℚ(t)) to arguments t ∈ \overline{ℚ}. (This includes conditions (i) and (ii) since the arguments of functional equations of P_{m} automatically satisfy (∗_{m}) and since replacing t ∈ \overline{ℚ} by t^{σ} replaces each value x_{i} = ϕ_{i}(t) by x_{i}^{σ}.) This conjecture, which is discussed in more detail in §10 of [Z], contains as a special case Milnor’s conjecture [M] that the only linear relations over ℚ of the Clausen function Cl_{2}(θ) = ∑_{n=1}^{∞} sin(nθ)/n^{2} at arguments θ ∈ ℚπ are those arising from the distribution relations

∑_{m=1}^{|N|} Cl_{2}(θ + \frac{2mπ}{N} ) = \frac{1}{N} Cl_{2}(Nθ) (0 ≠ N ∈ ℤ).

The only evidence in its support is its naturalness and internal consistency and the fact that the many known examples of algebraic relations among polylogarithms all conform to it. On the theoretical level nothing is known; so far as I know, for instance, one cannot prove that there is even a single pair of values D(x), D(y) with x and y algebraic which are linearly independent over ℚ, or even that there is a single value of D(x) which is not a rational number! The conjecture is also particularly daring because we know from §5 that there are linear dependences among values of the mth polylogarithm function for arbitrarily large values of m, but for m > 6 do not know whether these functions have any non-trivial functional equations at all.

7. Functional equations. The basic fact here, already mentioned in §1, is that the only requirement for a functional equation is the algebraic condition (∗_{m}), i.e., that for any combination ∑ n_{i}[x_{i}] of functions (of one or several variables t) x_{i}(t) which satisfies (∗_{m}) the corresponding sum of polylogarithms ∑ n_{i}P_{m}(x_{i}(t)) is independent of t. In this section we discuss how this criterion can be used to check known functional equations and to find new ones in an algorithmic way. We will express all functional equations in terms of P_{m} rather than Li_{m} in order to have well-defined values for complex arguments and to avoid lower-order terms in the equations; for real values of the arguments, the function L_{m} would do just as well because of the relation between L_{m} and P_{m} mentioned at the end of §1. A striking property of many of the functional equations is their high level of symmetry; we will emphasize this aspect in our discussion.

Example 1. The 5-term relation for the dilogarithm. The basic functional equation of the dilogarithm function is the 5-term relation (cf. Chapter I, section 1.2), which in our notation says that D(R_{5}(x, y)) = 0 for any x and y, where R_{5}(x, y) denotes the formal linear combination

(1)  R_{5}(x, y) = [x] + [y] + [\frac{1 − x}{1 − xy} ] + [1 − xy] + [\frac{1 − y}{1 − xy} ]

of elements in ℚ(x, y). (The other forms given by Spence, Hill, Abel and Kummer are equivalent to this by the 1-variable relations of Chapter I, 1.1.) To prove it in our language, we have to check that R_{5}(x, y) is in the kernel of the map β_{2} : [z] ↦ [z] ∧ [1 − z]. We calculate

β_{2}(R_{5}(x, y)) = [x] ∧ [1 − x] + [y] ∧ [1 − y] + ([1 − x] − [1 − xy]) ∧ ([x] + [1 − y] − [1 − xy])
+ [1 − xy] ∧ ([x] + [y]) + ([1 − y] − [1 − xy]) ∧ ([y] + [1 − x] − [1 − xy]) = 0.

This is the simplest example of the use of the calculus with wedge products in checking functional equations (and also, as far as the dilogarithm is concerned, the basic example, since it is apparently the case that all functional equations of the dilogarithm are consequences of the 5-term relation.)

We use this simple example to give a first illustration of the comments on symmetry made at the beginning of the section. First of all, we can check that R_{5} has a cyclic symmetry, i.e.,

R_{5}(x, y) = R_{5}(y, \frac{1 − x}{1 − xy} ) = R_{5}(\frac{1 − x}{1 − xy} , 1 − xy) = R_{5}(1 − xy, \frac{1 − y}{1 − xy} ) = R_{5}(\frac{1 − y}{1 − xy} , x).

However, this is only part of the full symmetry group. The basic invariance property of the dilogarithm is that for any x the six numbers D(x), D(1/x), D(1−1/x), D(x/(x−1)), D(1/(1−x)) and D(1−x) are equal up to sign. This 6-fold symmetry plays a role so often in the following that we introduce the special notation

x′ ∼_{6} x ⇔ x′ ∈ \{x, \frac{1}{x}, 1 − \frac{1}{x}, \frac{x}{x − 1}, \frac{1}{1 − x}, 1 − x\} .

A symmetric way to express the ∼_{6}-invariance is to say that the function D([a, b, c, d]), where

[a, b, c, d] = \frac{a − c}{a − d} \frac{b − d}{b − c} denotes the cross-ratio of a, b, c and d, is up to sign a symmetric function of its four arguments, since changing the order of four numbers replaces their cross-ratio x by a number x′ ∼_{6} x. More precisely,

(2)  D([x_{π(1)}, x_{π(2)}, x_{π(3)}, x_{π(4)}]) = sgn(π) D([x_{1}, x_{2}, x_{3}, x_{4}])

for any x_{i} ∈ ℙ^{1}(ℂ) (1 ≤ i ≤ 4) and any π ∈ 𝔖_{4}, the symmetric group on 4 letters. To verify this in our terminology we compute

β_{2}([x_{1}, x_{2}, x_{3}, x_{4}]) = [\frac{(x_{1} − x_{3})(x_{2} − x_{4})}{(x_{1} − x_{4})(x_{2} − x_{3})}] ∧ [\frac{(x_{1} − x_{2})(x_{3} − x_{4})}{(x_{1} − x_{4})(x_{2} − x_{3})}]

= \frac{1}{2} ∑_{π∈𝔖_{4}} sgn(π) [x_{π(1)} − x_{π(2)}] ∧ [x_{π(2)} − x_{π(3)}]

(this has only 12 terms rather than 24, because the πth summand is invariant under π ↦ πτ where τ is 1 ↔ 3). This has the desired invariance property under the action of 𝔖_{4}, so D([x_{1}, x_{2}, x_{3}, x_{4}]) also does. Now the 5-term relation becomes simply

(3) \frac{1}{24} ∑_{π∈𝔖_{5}} sgn(π) D([x_{π(1)}, x_{π(2)}, x_{π(3)}, x_{π(4)}]) = 0,  (x_{1}, …, x_{5} ∈ ℙ^{1}(ℂ))

(this has only 5 terms rather than 120 because the πth summand is unchanged by π ↦ πτ, τ ∈ 𝔖_{4}), with a symmetry group of order 120.

Example 2. The 9-term relation for the dilogarithm. As a second example, we consider the 3-variable functional equation (2.40) of Chapter I, 2.3.2. This equation has an obvious 8-fold symmetry generated by the involutions x ↔ y, v ↔ w and (x, y) ↔ (v, w), but in fact has a symmetry group of order 72, a typical non-obvious symmetry being given by (x, y, v, w) ↦ (y, xy/vw, y/w, y/v). To make the symmetries obvious, we first define a 3 × 3 matrix

(4)  Z = (z_{ij})_{i,j=1,2,3} = ( \begin{matrix}x & 1/v & v/x \\ 1/w & y & w/y \\ w/x & v/y & xy/vw \end{matrix} ) .

Then the functional equation becomes simply ∑_{i,j} D(z_{ij}) = 0. The constraints on the z_{ij} are

(5)  ∏_{j} z_{ij} = 1 (∀i) ,  ∏_{i} z_{ij} = 1 (∀j)

and (taking the indices i and j modulo 3)

(6)  (1 − z_{i,j})(1 − z_{i+1,j+1}) = (1 − z_{i,j+1}^{−1})(1 − z_{i+1,j}^{−1})  (∀i, j) .

Indeed, (5) is equivalent to the fact that Z can be expressed by (4) for some numbers x, y, v and w, and (6) for i = j = 1 is just the constraint (1 − x)(1 − y) = (1 − v)(1 − w). A calculation shows that this equation implies the validity of (6) for the other 8 values of the indices (i, j). Thus we have the symmetry group of order 72 given by permuting the indices i, permuting the indices j, and interchanging the roles of i and j. However, this is still not satisfactory because we cannot yet “see” why equation (6) for one pair of indices (i, j) implies (in conjunction with (5)) its truth for all i, j. To remedy this, we choose new coordinates t_{ij} (i, j ∈ ℤ/3ℤ) and set z_{ij} = t_{i,j} t_{i,j+1}^{−1} t_{i+1,j}^{−1} t_{i+1,j+1} . Then the equations (5) are true identically and each of the 9 equations (6) is equivalent to the condition det(T) = 0, where T = (t_{ij}). (The set of 3 × 3 matrices T with det(T) = 0 is 8-dimensional, but there are only 3 free parameters because replacing t_{ij} by λ_{i}μ_{j}t_{ij} for any λ_{i} and μ_{j} leaves z_{ij} unchanged.) Finally, we give a parametric solution of det(T) = 0 by setting t_{ij} = u_{i} −v_{j}. Then the constraints (5) and (6) are automatically satisfied and z_{ij} is simply the cross-ratio of the four numbers u_{i}, u_{i+1}, v_{j} and v_{j+1}. Therefore the functional equation can be written

∑_{i, j} D([u_{i}, u_{i+1}, v_{j}, v_{j+1}]) = 0 for any 6 points u_{i}, v_{j} ∈ ℙ^{1}(ℂ) (i, j ∈ ℤ/3ℤ).

Now not only the 72-fold symmetry of the equation is obvious (permute the u_{i}’s or the v_{j}’s or interchange u and v), but also its proof: applying (3) to the 5-tuple (u_{i}, u_{i+1}, v_{1}, v_{2}, v_{3}) gives

∑_{j} D([u_{i}, u_{i+1}, v_{j}, v_{j+1}]) = D([u_{i}, v_{1}, v_{2}, v_{3}]) − D([u_{i+1}, v_{1}, v_{2}, v_{3}])

for each i, and the sum of this over i (mod 3) vanishes. Of course, we can now go back and give a non-motivated and non-symmetric proof of the equation in its original form by adding the three 5-term relations R_{5}(x, 1/v), R_{5}(1/w, y) and R_{5}(w/x, v/y) and using the 1-variable functional equations of the dilogarithm to simplify the result.

Example 3. The 9-term relation for the trilogarithm. As the first example of the verification of a functional equation for a higher-order polylogarithm we consider Kummer’s 2-variable equation for the trilogarithm. Again we use P_{3} rather than Li_{3} in order to eliminate lower-order terms (the function L_{3} would do just as well if we restricted our attention to real values of the variables). The functional equation in question (cf. Chapter 2, 3.2.2.3) has the form P_{3}(R_{9}(x, y)) = 2ζ(3), where R_{9}(x, y) denotes the element

R_{9}(x, y) =2[x] + 2[y] + 2[\frac{x(1 − y)}{x − 1}] + 2[\frac{y(1 − x)}{y − 1}] + 2[\frac{1 − x}{1 − y}] + 2[\frac{x(1 − y)}{y(1 − x)}]

− [xy] − [\frac{x}{y}] − [\frac{x(1 − y)^{2}}{y(1 − x)^{2}} ] (7)

of ℚ(x, y). This equation has an obvious symmetry group of order 8 (invert x or y or interchange x and y), but in fact has a symmetry group of order 24. We can write it symmetrically in two ways.

Symmetric form: first version. Let H be the hypersurface in ℙ^{3}(ℂ) consisting of 4tuples (a_{1} : a_{2} : a_{3} : a_{4}) with ∑_{i} a_{i} = 0, with the obvious action of the group 𝔖_{4}. Let V be the 3-dimensional space generated multiplicatively by all ratios a_{i}/a_{j} and W ⊃ V the 6-dimensional space generated by all quotients (a_{i} +a_{j})/a_{i}, j ≠ i (this space has dimension only 6 because up to sign there are only three distinct quantities a_{i} + a_{j}). Each of the elements x = −a_{i}/a_{j} (j ≠ i) and a_{i}a_{j}/a_{k}a_{l} (i, j, k, l distinct) has the property that x ∈ V and 1 − x ∈ W, and up to inversion there are exactly 9 of them. The argument R_{9} of Kummer’s functional equation can then be written symmetrically as

2 ( [−\frac{a_{1}}{a_{2}} ] + 5 permutations ) − ( [\frac{a_{1}a_{2}}{a_{3}a_{4}} ] + 2 permutations ) .

(To see this, normalize the common value of a_{1} + a_{2} and −a_{3} − a_{4} to be 1, in which case H is parametrized by two parameters x = −\frac{a_{1}}{a_{2}} , y = −\frac{a_{3}}{a_{4}} as \{(\frac{x}{x−1} : \frac{1}{1−x} : \frac{y}{1−y} : \frac{1}{y−1})\}.) The proof of P_{3}(R_{9}) = constant (the value of the constant is then found by specializing to x = 0, y = 1) can be obtained easily by computing β_{3}(−a_{1}/a_{2}) and β_{3}(a_{1}a_{2}/a_{3}a_{4}) and symmetrizing with respect to the group action.

Symmetric form: second version. As with the dilogarithm, we interpret the arguments of the trilogarithms as cross-ratios of 4-tuples of points in ℙ^{1}(ℂ). However, we no longer have the sixfold symmetry of the function D(x), but only the twofold symmetry P_{3}(x) = P_{3}(1/x). This means that the argument of P_{3}(x) must be interpreted as a cross-ratio [a, b, c, d] of four points a, b, c, d ∈ ℙ^{1}(ℂ) where the only symmetries allowed are the ones generated by the interchanges of a and b, of c and d, or of (a, b) and (c, d), i.e. P_{3}([a, b, c, d]) depends only on the unordered pair of unordered pairs {{a, b}, {c, d}}. To emphasize this, we write the cross-ratio x as [a, b; c, d] rather than [a, b, c, d], although the definition is the same as before. Now suppose that we have six points p_{ν} ∈ ℙ^{1}(ℂ) and that there is an involution τ on ℙ^{1}(ℂ) which interchanges these points in pairs. We consider all cross-ratios [a, b; c, d] of four distinct points a, b, c, d from the set {p_{ν}} for which {a, b} is disjoint from {τ(c), τ(d)}. Using the symmetries of [a, b; c, d] and the invariance of the cross-ratio under automorphisms of ℙ^{1}(ℂ), we find that there are 9 of these, 3 of the form [a, τ(a); c, τ(c)] and 6 of the form [a, τ(a); c, d] with d ≠ τ(c). Then R_{9} is just twice the sum of the latter 6 minus the sum of the former 3. (Take the p_{ν} to be 0, ∞, 1, xy, x and y with the involution τ : t ↦ xy/t.) The equivalence of this form of the functional equation to the one just given is seen by taking for {p_{ν}} the 6 numbers a_{i} + a_{j} (1 ≤ i < j ≤ 4) and for τ the involution t ↦ −t. The symmetry group appears now not as 𝔖_{4}, but as the semi-direct product of 𝔖_{3} (permute the three orbits of τ on {p_{ν}}) with a Klein 4-group (interchanging the two elements within each orbit is an automorphism of order 2, but the product of these three automorphisms is just τ and has no effect on the cross-ratio).

The functional equations discussed up to now are classical, but the method of verifying them can also be used to discover new equations. The basic desideratum, as in the case of relations among special values, is to find as many x as possible such that all the x and 1 − x belong to subspaces V and W of F^{×} of small dimension. Typically one first chooses the generators of V (certain irreducible polynomials in one or several variables), preferably with a lot of symmetry to reduce the number of independent conditions which have to be checked later, and then looks for many x in V for which the prime factors of the elements 1 − x are repeated many times; then one computes β_{m}(x) for these x’s and some small m and uses linear algebra to solve if possible the system of equations which expresses that a linear combination of them vanishes. This algorithm, which can be carried out by hand in simple cases, can also be programmed, although not easily. This has been carried out by H. Gangl, who in this way found a large number of new functional equations. In the remainder of the appendix we present a few of these.

Example 4. Functional equations for the tetralogarithm. The simplest of Gangl’s functional equations is the 9-term equation

(8)  \begin{matrix}2 [P_{4}(z(1 − z)) + P_{4}(−\frac{z}{(z − 1)^{2}} ) + P_{4}(\frac{z − 1}{z^{2}})] \\ − & 3 [P_{4}(\frac{1}{1 − z + z^{2}} ) + P_{4}(\frac{(1 − z)^{2}}{1 − z + z^{2}} ) + P_{4}(\frac{z^{2}}{1 − z + z^{2}} )] \\ − & 6 [P_{4}(\frac{1 − z + z^{2}}{z(z − 1)} ) + P_{4}(\frac{1 − z + z^{2}}{z} ) + P_{4}(\frac{1 − z + z^{2}}{1 − z} )] = 0 . \end{matrix}

This has a 6-fold symmetry G_{9}(z′) = G_{9}(z) for z′ ∼_{6} z and up to this symmetry has only three rather than nine terms, as shown by the square brackets.

More interesting is the equation

[P_{4}(−\frac{bd}{a}) + P_{4}(−\frac{ac}{d}) + P_{4}(−\frac{ab^{3}}{c^{2}d}) + P_{4}(−\frac{c^{3}d}{ab^{2}}) + P_{4}(−\frac{d}{ab^{2}} ) + P_{4}(−\frac{a}{c^{2}d})]

+ [P_{4}(−\frac{ad^{2}}{b} ) + P_{4}(−\frac{a^{2}d}{c})] + 2[P_{4}(\frac{b}{c^{2}} ) + P_{4}(\frac{c}{b^{2}} ) + P_{4}(bc)]

+ 3[P_{4}(−\frac{b}{ad}) + P_{4}(−\frac{c}{ad}) + P_{4}(−\frac{1}{ad})]

+ 3[P_{4}(−cd) + P_{4}(−ab) + P_{4}(−\frac{d}{c}) + P_{4}(−\frac{a}{b}) + P_{4}(−\frac{ab}{c^{2}}) + P_{4}(−\frac{cd}{b^{2}})]

+ 6 [P_{4}(\frac{c}{a}) + P_{4}(\frac{b}{d}) + P_{4}(\frac{1}{d}) + P_{4}(\frac{1}{a}) + P_{4}(\frac{b}{cd}) + P_{4}(\frac{c}{ab})] = 0 ,

in two variables y and z, where we have abbreviated

a = 1 − z + yz, b = −y, c = y − 1, d = 1 − y + yz

This equation has an obvious symmetry under the involution P : (a, b, c, d) ↦ (d, c, b, a), corresponding to (y, z) ↦ (1 − y, 1 − z), and a less obvious one under the involution Q : (a, b, c, d) ↦ (a, b^{−1}, cb^{−1}, db^{−1}), corresponding to (y, z) ↦ (y^{−1}, −yz). These two involutions generate a group of order 12 under which the 26 terms of the functional equations fall into only 6 orbits (grouped by square brackets in the formula above). One can introduce new variables which make the symmetries obvious. Let F be the function field (over ℚ) of the variety

X = {(t_{1} : t_{2} : t_{3} : t_{4} : t_{5}) ∈ ℙ^{4} | ∑_{i=1}^{5} t_{i} = 0, ∑_{i=1}^{3} t_{i}^{−1} = 0 } ,

with an obvious symmetry of the group G = 𝔖_{3} × 𝔖_{2} ⊂ 𝔖_{5}. We can parametrize X by

(t_{1} : t_{2} : t_{3} : t_{4} : t_{5}) = (−y : y − 1 : y(1 − y) : y − y(1 − y)z : (y − 1)^{2} + y(1 − y)z ) ,

(set y = t_{3}/t_{2}, z = −(t_{1} + t_{4})/t_{3}) and under this identification G is identified with the symmetry group ⟨P, Q⟩ of the functional equation. The 26 arguments x of P_{4} satisfy x ∈ V, 1 − x ∈ W where V is the 4-dimensional subspace of F^{×} ⊗_{ℤ} ℚ spanned by the quotients t_{i}/t_{j} (i ≠ j) and W the 11-dimensional space spanned by V and the elements (t_{i} + t_{j})/t_{i} (i ≠ j, max{i, j} ≥ 4). Using this description and the symmetry one can easily check that the linear combination of the x’s specified by the functional equation is in the kernel of β_{4}.

Example 5. A 2-variable functional equation for the hexalogarithm. Gangl found one-variable functional equations for the penta- and hexalogarithm which had a similar structure to equation (8) above, namely the arguments of the polylogarithms involve only the irreducible polynomials z, 1 − z and 1 − z + z^{2} and the whole equation is unchanged if z is replaced by any z′ ∼_{6} z. The functional equation for the hexalogarithm, in particular, has 60 terms forming 13 orbits under the 6-fold symmetry and can be written efficiently as

3T_{{0,1,5}}^{−} + 4T_{{−5,0,3}}^{+} + 5T_{{−4,3,3}}^{−} − 20T_{{0,1,3}}^{−} − 20T_{{0,0,4}}^{−} − 60T_{{−3,1,2}}^{−} + 90T_{{−1,1,2}}^{−}

+ 95T_{{−1,0,3}}^{−} + 120T_{{−2,1,1}}^{+} − 180T_{{−1,1,2}}^{+} − 360T_{{0,1,1}}^{+} + 540T_{{0,1,1}}^{−} − 544T_{{0,0,2}}^{+} = 0 ,

where for a set of three integers α, β, γ with even sum we have set

T_{\{α,β,γ\}}^{±} = ∑_{\{a,b,c\}=\{α,β,γ\}} P_{6}(±\frac{(−1)^{a}z^{b}(1 − z)^{c}}{(1 − z + z^{2})^{(a+b+c)/2}} )

(the sum is over all permutations and has 6 terms in general but only 3 if two of α, β and γ coincide). Later he found a second functional equation with 89 terms and where the prime factors of the hexalogarithm arguments involved four prime factors z, 1 − z, 1 + z and 1−z +z^{2}. Many of the coefficients in these two equations agreed, suggesting that they might be different specializations of the same two-variable equation. After a considerable amount of trial and error, guided by the symmetry, it was found that this is true and that the two-variable functional equation has an 18-fold symmetry:

Theorem. Let y and z be variables and denote by y_{i} (1 ≤ i ≤ 6) and w_{j} (1 ≤ j ≤ 3) the elements y′ ∼_{6} y and z′(1 − z′), z′ ∼_{6} z. Then

(9)  ∑_{a,b} n_{a,b} \frac{1}{μ_{a,b}} ∑_{i=1}^{6} ∑_{j=1}^{3} P_{6}((\frac{w_{j}}{y_{i}} )^{a}(\frac{1 − w_{j}}{1 − y_{i}} )^{b})

is independent of y, where the coefficients n_{a,b} and μ_{a,b} are given by the following table:

\begin{matrix}(a, b) & (−2, 3) & (−1, −1) & (−2, 1) & (1, −2) & (1, 0) & (0, 1) & (1, −1) \\ n_{a,b} & 3 & 4 & 5 & 20 & 60 & 90 & 180 \\ μ_{a,b} & 3 & 2 & 2 & 2 & 1 & 1 & 1 \end{matrix}

Remarks. 1. The value of (9) can be found by specializing y in any way we want. Taking y = ∞ (or 0 or 1, which are equivalent under ∼_{6}), we find that it equals 60P_{6}(G_{9}(z)),

where G_{9}(z) is the combination of arguments occurring in Gangl’s tetralogarithmic functional equation (8). 2.The coefficient of each inner double sum in (9) has been written as n_{a,b}/μ_{a,b} rather than simply n_{a,b} because the 18 terms of the double sums occur with multiplicity μ_{a,b}, so that the (a, b)th summand in (9) in fact consists of 18/μ_{a,b} terms with coefficient n_{a,b}. Thus there are 87 terms altogether, forming 7 orbits under a group of order 18, or 96 terms forming 8 orbits if we include the “constant term” as given in the previous remark.

3. Gangl’s original 1-variable equations are obtained by specializing to y = z and y = −z.

4. The coefficients n_{a,b} are determined by the requirement that the homogeneous polynomial ∑ n_{a,b}(aX + bY)^{5} vanish identically (since this polynomial is a combination of 6 monomials X^{k}Y^{5−k} and there are 7 indices (a, b), this has a solution). To see the necessity of this condition, note that the arguments x of the hexalogarithms in (9) belong to the 16-dimensional space generated by the irreducible polynomials y_{i}/w_{j} and (1 − y_{i})/(1 − w_{j}) (1 ≤ i ≤ 6, 1 ≤ j ≤ 3), while 1 − x belongs to the 34-dimensional space with the additional generators p_{i,j} = y_{i} − w_{j}. The contribution of one of the new primes p = p_{i,j} to β_{6}(x), where x is one of the arguments in (9), is ord_{p}(x) [x]^{5} ⊗ [p] ∈ Sym^{5}(F^{×}) ⊗ F^{×} (F = ℚ(y, z)), so a short consideration shows that the total coefficient of [p_{i,j}] under β_{6} equals ∑ n_{a,b}(aX + bY)^{5} with X = [y_{i}/w_{j}], Y = [(1 − y_{i})/(1 − w_{j})] ∈ F^{×}. Thus the functional equation stated in the theorem can only hold for the values of n_{a,b} given in the table; that it actually does hold depends on luck (at least at our present stage of understanding), since the coefficients are already uniquely determined by requiring that the “new” primes p_{i,j} drop out under the map β_{6}, and we have no further freedom to ensure that the “old” primes also give a zero contribution modulo Sym^{6}(F^{×}).

References

[AL]. M. Abouzahra and L. Lewin, The polylogarithm in algebraic number fields, J. Number Theory 21 (1985), 214–244.

[B].A. Borel, Cohomologie de SL_{n} et valeurs de fonctions zêta aux points entiers, Ann. Sc.

Norm. Sup. Pisa 4 (1977), 613–636. [BD]. A.A. Beilinson and P. Deligne, in preparation. [EST]. P. Erdös, C. Stewart and R. Tijdeman, Some diophantine equations with many solu-

tions, Comp. Math. 66 (1988), 36–56. [G].A.B. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology,

preprint, 1990. [L].L. Lewin, Polylogarithms and Associated Functions, North-Holland, New York, 1981. [M].J. Milnor, Hyperbolic geometry: the first 150 years, Bull AMS 6 (1982), 9–24. [S].A.A. Suslin, Algebraic K-theory of fields, in Proceedings of the International Congress

of Mathematicians 1986, AMS, 1987, pp. 222–244. [Z].D. Zagier, Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of

fields, in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston, 1991, 391–430.

Remark. For computing numerical examples in number fields it is useful to have a simple and rapidly convergent formula for calculating Li_{m}(z). For z small (say |z| < 1/2), one can of course simply use the defining series of the polylogarithm, and for z large the functional equation relating the polylogarithms of z and 1/z. For z near the unit circle, a convenient (and pretty) formula which was noticed by Henri Cohen and myself is

Li_{m}(e^{x}) = ∑_{n=0}^{∞} ζ(m − n) \frac{x^{n}}{n!} ,

where the meaningless term ζ(−1) is to be replaced by 1 + \frac{1}{2} + ⋯ + \frac{1}{m−1} − log(−x). This formula, easily proved by m-fold differentiation, works for all x of absolute value less than 2π and hence gives Li_{m}(z) for all z with .005 < |z| < 230.