FUNCTIONAL EQUATIONS FOR HIGHER LOGARITHMS

HERBERT GANGL

Contents

1. Introduction 1 2. A general dilogarithm equation 2 3. Trilogarithm equations 2 4. 4-logarithm equations 6 5. A 7-logarithm equation 7 6. Appendix: a proof of Theorem 4.1 9 References 12

Abstract. Following earlier work by Abel and others, Kummer gave functional equations for the polylogarithm function Li_{m}(z) up to m = 5 in 1850, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7–logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.

1. Introduction

1.1. An essential property of the logarithm is its functional equation

log(xy) = log(x) + log(y) .

An essential property of the dilogarithm, which is defined by the power series

Li_{2}(z) = ∑_{n>0} \frac{z^{n}}{n^{2}} , |z| < 1,

and can be analytically continued to C − [1, ∞) , is the 5-term equation (note the involutary symmetry (x, y) ↦ (\frac{1−x}{1−y^{−1}} , \frac{1−y}{1−x^{−1}} ))

Li_{2}(xy) − Li_{2}(x) − Li_{2}(y) − Li_{2}(\frac{1 − x}{1 − y^{−1}} ) − Li_{2}(\frac{1 − y}{1 − x^{−1}} ) = (elementary) .

Here the right hand side is a sum of products of logarithms.

A similar functional equation is expected for the higher polylogarithms

Li_{m}(z) = ∑_{n>0} \frac{z^{n}}{n^{m}} , |z| < 1.

The relation of polylogarithms and algebraic K−theory, and in particular Zagier’s conjecture on special values of Dedekind zeta functions, have made it clear that the question of finding functional equations for higher polylogarithms Li_{m} is a central one (cf. e.g. [10]). Kummer gave special functional equations for m ≤ 5 and Goncharov [3] found a basic one for m = 3 as the key step in his proof of Zagier’s conjecture for this case, but almost nothing is known in the general case. In our thesis [1] we gave the first functional equations for m = 6 and m = 7 as well as many new examples for m ≤ 5 , some of which are contained in the following.

1.2. Recall that the function Li_{m}(z) is a many-valued function on C−{0, 1} but that one has the one-valued continuous function

L_{m}(z) = ℜ_{m}( ∑_{r=0}^{m} \frac{2^{r}B_{r}}{r!} log^{r} |z| Li_{m−r}(z)), |z| ≤ 1, z ∉ \{0, 1\},

where ℜ_{m} denotes the real part for m odd and the imaginary part for m even, B_{r} the r-th Bernoulli number ( B_{0} = 1, B_{1} = −1/2, B_{2} = 1/6, …). For |z| > 1 , L_{m}(z) is given by the functional equation L_{m}(z) = (−1)^{m−1}L_{m}(1/z) , while for z = 0 and z = 1 one extends the function by continuity. For m = 2 this is the famous Bloch–Wigner dilogarithm. For m > 2 one-valued versions of Li_{m} were introduced by Ramakrishnan, Wojtkowiak and Zagier. The above function is due to Zagier [9] who denoted it P_{m}(z) . It has the advantage that the corresponding functional equations of Li_{m} become “clean” for L_{m}, e.g. in the 5-term equation stated above the elementary term on the right hand side disappears when we replace Li_{2} by L_{2} .

Note that we can consider L_{m} as a function on C[C] by extending it linearly.

2. A general dilogarithm equation

Let us first recall a general functional equation for the dilogarithm found by Rogers —for a polynomial ϕ without constant term—and generalized by Wechsung [6], Wojtkowiak [8] and Zagier [11] to any rational function ϕ. For m ∈ N put

L̃_{m}(x, y, z, w) = L_{m}(cr(x, y, z, w)) = L_{m}(\frac{x − z}{x − w} \frac{y − w}{y − z} ) ,

where cr denotes the cross ratio.

Proposition 2.1. Let ϕ : P_{C}^{1} → P_{C}^{1} be a rational function, α, B, C, D ∈ P^{1}(C) . Then, putting A = ϕ(α) , we obtain

∑_{β,γ,δ} L̃_{2}(α, β, γ, δ) = deg(ϕ) · L̃_{2}(A, B, C, D) ,

where β, γ and δ run through the preimages of B, C and D, respectively, with multiplicities.

The proposition contains as a special case the five term relation (e.g. if we take ϕ(z) = z(1 − z) and B = 1, C = 0, D = ∞) and in fact many other equations. A functional equation which is not covered by the proposition (and presumably the only one, essentially) is the one relating z and z̄, its complex conjugate, via L_{2}(z̄) = −L_{2}(z).

3. Trilogarithm equations

The following equation for the trilogarithm was found by symmetrizing a functional equation given by Wojtkowiak [8] (which in a similar form had been given earlier by Wechsung, cf. [6], [7]).

Proposition 3.1. Let ϕ : P_{C}^{1} → P_{C}^{1} be a rational function, A_{i}, B_{j}, C_{k}, D_{l} ∈ P^{1}(C) , i, j, k, l ∈ {1, 2}. Then

∑_{i,j,k,l} (−1)^{i+j+k+l}( ∑_{α_{i},β_{j},γ_{k},δ_{l}} L̃_{3}(α, β, γ, δ) − deg(ϕ) · L̃_{3}(A_{i}, B_{j}, C_{k}, D_{l})) = 0 ,

where α_{i}, β_{j}, γ_{k} and δ_{l} run through the preimages of A_{i}, B_{j}, C_{k} and D_{l}, respectively, with multiplicities.

Wojtkowiak’s original equation [8], pp.226-227, is obtained by specializing the equation from the proposition above and can be stated in a simpler form as follows:

Corollary 3.2. Let A, B, C ∈ P^{1}(C) , ϕ : P_{C}^{1} → P_{C}^{1} a rational function of degree n and x an independent variable. Let ϕ^{−1}(A) = {α_{i}}_{i=1}^{n}, ϕ^{−1}(B) = {β_{i}}_{i=1}^{n} and ϕ^{−1}(C) = {γ_{i}}_{i=1}^{n}. Then

L̃_{3} (ϕ(x), C, B, A) − ∑_{i,j,k=1}^{n} L̃_{3} (x, γ_{i}, β_{j}, α_{k})

− ∑_{i,j,k=1}^{n} L̃_{3} (x, α_{i}, α_{j}, γ_{k}) − ∑_{i,j,k=1}^{n} L̃_{3} (x, β_{i}, β_{j}, γ_{k})

+ ∑_{i,j,k=1}^{n} L̃_{3} (x, α_{i}, α_{j}, β_{k}) + ∑_{i,j,k=1}^{n} L̃_{3} (x, β_{i}, β_{j}, α_{k})

= const,

i.e. the expression on the left hand side is independent of x.

Here it is understood that α_{i}, β_{j} and γ_{k} run through all preimages (counted with multiplicity) of A, B and C, respectively.

3.1. Around Goncharov’s equation.

3.1.1. The original description. Goncharov [3] found a beautiful interpretation of functional equations in terms of configuration spaces, and as a by-product he provided an equation for the trilogarithm in three variables α_{i} for which he gave a threefold symmetry. We reproduce it here using the shorthand β_{i} = 1−α_{i}+α_{i}α_{i−1}, i = 1, 2, 3, where indices are understood modulo 3: put

(3.1) γ(α_{1}, α_{2}, α_{3}) =∑_{i=1}^{3}([\frac{1}{α_{i}}] + [β_{i}] + [\frac{α_{i}α_{i−1}}{β_{i}}] + [\frac{β_{i}}{β_{i+1}α_{i+2}} ] + [ − \frac{β_{i}α_{i+1}}{β_{i+1}} ])

+ [ − \frac{1}{α_{1}α_{2}α_{3}} ] − ∑_{i=1}^{3} ([\frac{β_{i}}{α_{i−1}} ] + [\frac{β_{i}}{β_{i+1}α_{i}α_{i−1}} ] + [1]) . (3.2)

Proposition 3.3. With the above notation,

L_{3}(γ(α_{1}, α_{2}, α_{3})) = 0 .

3.1.2. A more symmetric description. Since Goncharov’s equation plays such a central role for the theory, it seems worthwhile to analyze its structure a bit further. There is actually a much bigger symmetry group G (of order 192) than the cyclic group on three letters acting on the set of arguments and dividing the 22 terms into 2 orbits, one of length 16 (corresponding to the 15 terms in the first sum plus the single term [− \frac{1}{α_{1}α_{2}α_{3}} ]), the other one of length 6 (corresponding to the second sum, with the exclusion of the term [1]). G is generated by two involutions

π_{1} : (α_{1}, α_{2}, α_{3}) ↦ (α_{1}, α_{2}, −\frac{β_{1}}{α_{1}β_{3}} ) , π_{2} : (α_{1}, α_{2}, α_{3}) ↦ (\frac{1}{α_{1}} , \frac{1}{α_{3}} , \frac{1}{α_{2}} ) ,

together with the obvious symmetry of order 3 (shifting the indices mod 3). The set consisting of the arguments in the sixteen terms of the first orbit is transformed either into itself (e.g. via π_{1}) or into the set of inverses (e.g. via π_{2}). We give a presentation with more obvious symmetries: let t_{1}, …, t_{4} be four variables, subject to the constraint ∏_{i} t_{i} = 1 . Then the following element in ℚ(t_{1}, t_{2}, t_{3}) is annihilated by L_{3} for each evaluation of t_{1}, …, t_{3} in ℂ:

∑_{i} [t_{i}] + ∑_{{i,j \atop i≠j}} [\frac{1 − t_{i}}{1 − t_{j}^{−1}} ] − \frac{1}{4} ∑_{{i,j \atop i≠j}} [t_{i} t_{j}] − \frac{1}{8} ∑_{\{i,j,k,l\}=\{1,2,3,4\}} [\frac{(1 − t_{i})(1 − t_{j})}{(1 − t_{k}^{−1})(1 − t_{l}^{−1})}] −3[1] .

The obvious S_{4} -action on the set of arguments (where we identify [z] and [1/z]), together with the involution

t_{i} ↦ \frac{1 − t_{i}}{1 − t_{i+2}^{−1}} , i mod 4 ,

generates the symmetry group G (of order 192) mentioned above.

3.1.3. A yet more symmetric description. Consider the finite group G′ (of order 96) of automorphisms of ℚ(y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3}), generated by G′ = ⟨g, h⟩, where

(3.3)  g : (y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3}) ↦ (\frac{1}{y_{1}} , z_{2}, z_{3}, z_{1}, y_{2}, y_{3}),
(3.4)  h : (y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3}) ↦ (y_{2}, y_{3}, y_{1}, z_{2}, z_{3}, z_{1}) .

The orbits of y_{1} and of \frac{y_{1} − z_{3}}{1 − y_{1}z_{2}} \frac{y_{2} − z_{1}}{1 − y_{2}z_{3}} \frac{y_{3} − z_{2}}{1 − y_{3}z_{1}} under G′ have order 12

and 32, respectively. As in the previous subsection, we introduce “parametrization variables” t_{1}, …, t_{4}, subject to the constraint ∏_{j=1}^{4} t_{j} = 1, and form 6 arguments

\{A_{i}\}_{i=1,2,3} = \{t_{i}t_{4}\}_{i} , \{B_{i}\}_{i=1,2,3} = \{\frac{1 − t_{j}^{−1}}{1 − t_{i}} \frac{1 − t_{k}^{−1}}{1 − t_{4}} | \{i, j, k\} = \{1, 2, 3\}, j < k\} .

The automorphism induced by

(t_{1}, t_{2}, t_{3}, t_{4}) ↦ (ι(t_{1}, t_{4}), ι(t_{4}, t_{1}), ι(t_{2}, t_{3}), ι(t_{3}, t_{2}))

where ι(x, y) = \frac{1 − y^{−1}}{1 − x}, acts on (A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}) like the automorphism g

in (3.3) above while the automorphism induced by (t_{1}, t_{2}, t_{3}) ↦ (t_{2}, t_{3}, t_{1}) acts like h in (3.4). If we put y_{i} resp. z_{i} equal to A_{i} resp. B_{i} in the above, the second orbit contains only 16 different terms up to inversion of the arguments—obviously, the first orbit contains only 6 terms up to inversion—and the resulting 16+6 arguments coincide precisely with the ones occurring in Goncharov’s equation.

3.1.4. Other descriptions. A different way to use the above parametrization in order to exhibit the symmetries among the 22 terms is obtained by putting α_{i} = −\sqrt{A_{i}}, β_{i} = \sqrt{B_{i}}. Then the four elements α_{1}^{ε_{1}}α_{2}^{ε_{2}}α_{3}^{ε_{3}} with ε_{j} = \{±1\} and ∏ ε_{j} = −1, together with the elements β_{1}^{ε_{1}}β_{2}^{ε_{2}}β_{3}^{ε_{3}} with ε_{j} = {±1} and ∏ ε_{j} = 1, form the 16 arguments in the second orbit above, while the α_{i}^{2} and β_{i}^{2} give the 6 arguments in the first one.

Yet another description in terms of equations instead of parametrizations is given by the following 9 (redundant) equations where we put q(x, y) = \frac{x − y}{1 − xy}:

B_{i}^{−1} = \frac{q(B_{i−1}, B_{i+1})}{q(A_{i+1}, A_{i−1})} ,

A_{i}^{−1} = q(A_{i+1}, \frac{1}{B_{i−1}} ) q(A_{i−1}, \frac{1}{B_{i+1}} ) ,

\frac{A_{i+1}(1 − A_{i})^{2}}{A_{i}(1 − A_{i+1})^{2}} = \frac{B_{i+1}(1 − B_{i})^{2}}{B_{i}(1 − B_{i+1})^{2}} ,

where i = 1, 2, 3 and the indices are taken modulo 3. (Obviously, we can break the symmetry and reduce this to a system of 7 equations in only 4 variables by eliminating, say, A_{1} and B_{1}. The system of equations is no complete intersection, though.)

3.2. Relating Goncharov’s and Wojtkowiak’s equations. In order to compare the equations resulting from Goncharov’s and Wojtkowiak’s approach, we propose to study an “intermediate” relation, with 34 terms, which allows an interpretation for both situations.

1. In Wojtkowiak’s equation, the “intermediate” relation occurs when we consider the rational function

ϕ(x) = \frac{x − a}{x − c^{−1}} · \frac{x − b}{x − abc} = cr(x, 0, a, c^{−1}) cr(x, 0, b, abc) ,

(for x = 1 this is a B_{i}-type term above) and (A, B, C) = (∞, 0, 1), and subtracting two different specializations for x of the ensuing terms from (3.2)—each specialization providing 17 terms—we are left with 34 terms which form a functional equation in three variables. The generic 17 terms are given, in factored form, as follows:

[\frac{(1 − ct)a}{a − t}] + [\frac{(1 − ct)b}{b − t}] + [\frac{(1 − ct)c}{t − a}] + [\frac{(1 − ct)c}{t − b}]

+[\frac{abc − t}{(a − t)bc}] + [\frac{abc − t}{(b − t)ac}] + [\frac{abc − t}{a − t}] + [\frac{abc − t}{b − t}]
+[\frac{(a − t)b(ac − 1)}{(b − t)a(bc − 1)}] + [\frac{(t − a)(1 − bc)}{(t − b)(1 − ac)}] + [\frac{(tc − 1)b(ac − 1)}{(abc − t)(bc − 1)}] + [\frac{(tc − 1)a(bc − 1)}{abc − t)(ac − 1)}]
−[\frac{(1 − ct)abc}{abc − t}] − [\frac{(1 − ct)}{(abc − t)c}] − [\frac{t − a}{t − b}] − [\frac{b(a − t)}{a(b − t)}] − [\frac{(b − t)(a − t)c}{(abc − t)(1 − ct)}] .

2. Let us recall that Goncharov [3] has defined a triple ratio for 6 points in the projective plane in such a way that it respects a 7-term relation: for any 7 distinct points P_{1}, …, P_{7} in ℙ^{2} one has

∑_{i=1}^{7} triple ratio(P_{1}, …, P̂_{i}, …, P_{7}) = 0 .

In the setting of configurations, the intermediate 34-term relation mentioned above encodes the well-definedness of the triple ratio: take a configuration of 6 points P_{i}, i = 1, …, 6, in the plane where P_{1}, P_{2} and P_{3} lie on a line:

[figure: •; •; •; P_{1}; P_{2}; P_{3}; •; Q; •; P_{6}; •; P_{5}; •; P_{4}; ℓ]

Goncharov reduces this configuration with the help of the 7-term relation to more degenerate configurations by introducing the intersection point Q of the line through the three points P_{1}, P_{2}, P_{3} and a line ℓ passing through 2 of the other 3 points (here P_{5} and P_{6}). He uses the fact that he has already defined the triple ratio for the more degenerate configurations which result by leaving out any other of the 6 original points. But there are different possibilities for choosing ℓ—e.g. one could take ℓ to be the line through P_{4} and P_{5}—and one verifies (this is not explicitly done in [3]) that the difference of two such possibilities essentially results in two sets of 17 terms which correspond precisely to the 34 terms in question. (Here “essentially” alludes to the fact that we argue “modulo” the inversion relation L_{3}(x) = L_{3}(1/x) and the 3-term relation L_{3}(x) + L_{3}(1/(1 − x)) + L_{3}(1 − 1/x) = L_{3}(1).)

3.3. Relating the 34-term and 22-term equation. If we take the difference of the specializations of the terms in (3.2) to x = 1 and x = 0, respectively, then a number of terms degenerate and we are left with the sum of a Kummer-Spence relation and a 22-term relation. (Again, we work up to inversion and 3term relation.) Similarly, the difference of the specializations to x = a and x = 0 give a version of the Kummer-Spence relation itself. Conversely, the following transformation (t_{1}, t_{2}, t_{3}) ↦ (cr(t, 0, c^{−1}, a), cr(t, 0, b, c^{−1}), cr(t, 0, a, abc)) maps the terms A_{i}, B_{i}, i = 1, 2, 3, in the 22-term relation above to the “generic 17 terms” as in 3.2.1 together with five terms which are independent of t. This shows immediately that the 22-term relation implies the 34-term relation.

3.4. A 21-term equation. Yet another related equation of possible interest is a sum of four 22-term relations which combines to a functional equation in 3 variables but with less, viz. only 21, different arguments (up to inversion), but with coefficients in {±1, ±2}: with the notation in 3.1, consider

Γ(x, y, z) = γ(\frac{1}{1 − x}, \frac{1 − x}{1 − xy} , 1 − z) + γ(1 − \frac{1}{x}, \frac{1 − xy}{y(1 − x)}, \frac{1}{1 − z^{−1}} )

and note that its symmetrization in the first 2 arguments has the following form— we introduce a “symmetrizing variable” z_{2} = (x_{1}x_{2}z_{1})^{−1} and j(t, u) = \frac{1−u^{−1}}{1−t} :

Γ(x_{1}, x_{2}, z_{1})+Γ(x_{2}, x_{1}, z_{1}) = −2[x_{1} x_{2}] − 2[1]
+2 ∑_{i=1}^{2} ([x_{i}] + [j(x_{i}, x_{3−i})] + [z_{i}] + [j(z_{i}, z_{3−i})]
−[x_{i} j(z_{1}, z_{2})] − [j(x_{i}, x_{3−i}) j(z_{1}, z_{2})] − [x_{i}z_{1}] − [j(x_{i}, x_{3−i}) z_{1}])
+ ∑_{i=1}^{2} ([x_{i} z_{1} j(x_{i}, x_{3−i}) j(z_{1}, z_{2})] + [x_{i} z_{1} / j(x_{i}, x_{3−i}) j(z_{1}, z_{2})]) .

4. 4-logarithm equations

It is in general a tedious job to verify even a single functional equation for the polylogarithms. Therefore it seems worthwhile finding a rather short (and elementary) way to verify a whole family of them. We propose to do this for a family of equations for the 4-logarithm, using essentially only one (and actually trivially checked) polynomial equation. One can proceed similarly (albeit in a somewhat more complicated fashion) with other families (cf. [2], Thm 4.4 and Thm. 4.9), even up to n = 6. Let us emphasize that not only had there been no example for n ≥ 6 previously known (apart from the trivial distribution relations) but Wechsung [6] even proved that the type of equation (called “Kummer-type”) which gave the only examples for n = 4, 5 is not “good enough” for n ≥ 6. The following example is not of Kummer-type. For a proof see the appendix.

Theorem 4.1. Let F be a field of characteristic 0, ϕ(x) = x^{n−1}(x − 1) for some n ∈ N. For some t, u ∈ F let {p_{i}}_{i=1}^{n} = ϕ^{−1}(t) resp. {q_{i}}_{i=1}^{n} = ϕ^{−1}(u) be the sets of preimages in some finite extension field F′ of F. Then the following element in Z[F′] is annihilated by L_{4}:

(4.1) n(n − 2)[\frac{∏_{i} p_{i}}{∏_{j} q_{j}} ] − (n − 1)^{2} ∑_{i,j=1}^{n} [\frac{1 − p_{i}^{−1}}{1 − q_{j}^{−1}} ] + n^{2} ∑_{i,j=1}^{n} [\frac{1 − p_{i}}{1 − q_{j}} ]

− n^{2}(n − 1)^{2} ∑_{i,j=1}^{n} [\frac{p_{i}}{q_{j}} ] + n(n − 1)^{2} ∑_{i=1}^{n} ([1 − \frac{1}{p_{i}} ] − [1 − \frac{1}{q_{i}} ]) .

5. A 7-logarithm equation

In this section we give a 2-variable functional equation for the 7–logarithm. This functional equation consists of 274 terms, but by using an appropriately symmetrized notation we can write it in a more digestible form.

5.1. Let F be the function field Q(t, u) . Let

f_{1}(z) = \frac{−z}{1 − z + z^{2}} , f_{2}(z) = \frac{z − 1}{1 − z + z^{2}} , f_{3}(z) = \frac{z(1 − z)}{1 − z + z^{2}}

and f(z) = −f_{1}(z)f_{2}(z)f_{3}(z) = \frac{z^{2}(1 − z)^{2}}{(1 − z + z^{2})^{3}} .

Notice that the group G ≅ S_{3} generated by z → 1/z and z → 1 − z permutes the f_{i} and leaves f invariant. For a, b, c, d ∈ Z define elements {a, b; c, d}_{0}(t, u) and {a, b; c, d} in C[C] by

\{a, b; c, d\}_{0}(t, u) = ∑_{i,j=1}^{3} [\frac{f(t)^{a}f_{i}(t)^{b−a}}{f(u)^{c}f_{j}(u)^{d−c}} ] and

{a, b; c, d} = {a, b; c, d}_{0}(t, u) + {c, d; a, b}_{0}(t, u)
= {a, b; c, d}_{0}(t, u) + {a, b; c, d}_{0}(u, t) ,

where the latter equation holds only up to inversions of the arguments (which, of course, does not affect the functional equation to be presented below). Finally, set

ξ_{7}^{(3)} = − \frac{1}{18} · \frac{609}{4} \{−1,−1;−1,−1\} − \frac{1}{3} · 35 \{−1,−1;−2, 1\} + \frac{1}{3} · \frac{105}{8} \{−1,−1; 3,−5\}
− \frac{1}{3} · 21 \{−1,−1;−1, 4\} − \frac{1}{3} · 15 \{−1,−1;−2, 5\} + \frac{1}{3} · 15 \{−1,−1; 3,−4\}
ξ_{7}^{(2)} = + \frac{1}{2} · 700 \{ 1, 0; 1, 0\} + \frac{1}{2} · \frac{175}{4} \{ 1,−3; 1,−3\} + \frac{1}{2} · 28 \{−2, 3;−2, 3\}
−  35 { 1,−3;−2, 3} − 140 {−2, 3; 1, 0} + 175 { 1, 0; 1,−3}
ξ_{7}^{(1)} = + \frac{1}{2} · 700 \{ 1,−2;−1, 2\} + 3150 \{ 0, 1; 1,−1\} + \frac{1}{2} · 1575 \{−1, 1; 1,−1\}
− 2100 \{ 1,−2; 0,−1\} + \frac{1}{2} · 6300 \{ 0, 1; 0,−1\} − 1050 \{−1, 2;−1, 1\}
− \frac{1}{2} · 700 \{−1, 2;−1, 2\} − \frac{1}{2} · 1575 \{−1, 1;−1, 1\} − \frac{1}{2} · 6300 \{ 0, 1; 0, 1\}
+ 1050 {−1, 2; 1,−1} + 2100 { 0,−1;−1, 2} − 3150 { 1,−1; 0,−1} .

Theorem 5.1. The element ξ_{7} = ξ_{7}^{(1)} +ξ_{7}^{(2)} +ξ_{7}^{(3)} ∈ Q[F] is a functional equation for L_{7} , i.e.

L_{7}(ξ_{7}(t, u)) = 0

for any t, u ∈ C where f_{j}(t), f_{j}(u) ∉ {0, ∞}, j = 1, 2, 3 .

Proof. Functional equations for polylogarithms can be characterized by an algebraic criterion (cf. [9], Proposition 1), so the proof can be reduced to checking this criterion for ξ_{7} which was done with a computer program. The statement applies, of course, to any functional equation; the difficulty lies in finding functional equations, not in checking their validity.

5.2. Remark. In ξ_{7} the denominator of the first factor of a coefficient (which we suppress if it is 1) gives the multiplicity of an argument in the sum {a, b; c, d} (up to inverse, i.e. we identify [z] and [1/z] ). We associate a weight to each {a, b; c, d} by wt({a, b; c, d}) = wt(a, b) in the following way: setting wt(a, b) = \frac{1}{2}(|a| + |a + b| + |2a + b|) we find that in all terms {a, b; c, d} of ξ_{7} we have wt(a, b) = wt(c, d) . This will subdivide the 24 terms into three blocks of weights 3, 2 and 1, as indicated in the formatting above, namely into the first 6, the second 6 and the final 12 terms. We can observe another operation on these blocks which even includes the coefficients. We want to explain this in more detail.

5.3. More symmetric form of ξ_{7} . Let (Z^{3})_{0} = {(a, b, c) ∈ Z^{3} | a + b + c = 0} and (Z^{3})_{1} = {(a, a, b) ∈ Z^{3}}. We consider the map

(5.1)  Θ : Z^{3} → Z^{3} ,
(5.2)  (a, b, c) ↦ (a, −b − c, b − a)

which sends (Z^{3})_{0} to (Z^{3})_{1} , and a second one (the f_{i} are taken as in (5.1))

(5.3)  Φ : Z^{3} → {ϕ : P_{C}^{1} → P_{C}^{1} | ϕ rational} ,
(5.4)  α = (α_{1}, α_{2}, α_{3}) ↦ (ϕ_{α} : z ↦ ( − f_{1}(z))^{α_{1}}f_{2}(z)^{α_{2}}f_{3}(z)^{α_{3}}) .

The symmetric group S_{3} operates on (Z^{3})_{0} and Z^{3} via permutation. For each k ∈ Z we have the element (k, −1, 1 − k) ∈ (Z^{3})_{0} . We set

A_{k} = Θ(S_{3} · (k, −1, 1 − k)) , k = 1, 2, 3.

Then we have

A_{1} = {±(1, 1, −2), ±(−1, −1, 1), ±(0, 0, 1)} ,
A_{2} = {(2, 2, −3), (−1, −1, 3), (−1, −1, 0)} ,
A_{3} = {(−1, −1, −1), (−1, −1, 4), (−2, −2, 5), (−2, −2, 1), (3, 3, −4), (3, 3, −5)} .

We will denote by δ the S_{3}-invariant element (−1, −1, −1) of A_{3} which will play a special role. Finally we define for α = (α_{1}, α_{2}, α_{3}) ≠ 0 a “weight”

ω(α) = \frac{1}{α_{1} − α_{3}} .

Then the theorem can be restated by saying that

60 ξ_{7} = ∑_{i=1}^{3} ξ^{(i)}

with

ξ^{(1)} = − \frac{29}{20}[\frac{ϕ_{δ}(t)}{ϕ_{δ}(u)}] + ∑_{α∈A_{3}\\{δ\}} ω(α) ∑_{σ∈S_{3}/S_{2}} ([\frac{ϕ_{σα}(t)}{ϕ_{δ}(u)}] + [\frac{ϕ_{δ}(t)}{ϕ_{σα}(u)}])

ξ^{(2)} = \frac{20}{3} ∑_{α,β∈A_{2}} ω(α)ω(β) ∑_{σ,τ∈S_{3}/S_{2}} [\frac{ϕ_{σα}(t)}{ϕ_{τβ}(u)}]

ξ^{(3)} = − 30 ∑_{α,β∈A_{1}} ω(α)ω(β) ∑_{σ,τ∈S_{3}/S_{2}} [\frac{ϕ_{σα}(t)}{ϕ_{τβ}(u)}] .

Altogether we get 1+30+81+162 = 274 different arguments (up to inverse) since in the last sum (over A_{1} ) the arguments associated to (α, β) and (−α, −β) are inverse to each other.

6. Appendix: a proof of Theorem 4.1

Proof. Using Zagier’s criterion (cf. [9], Proposition 1) alluded to earlier, we need only verify that the above combination lies in the kernel of the map

β_{4} : Z[F′] ⟶ Sym^{2}F′^{×} ⊗ ⋀^{2} F′^{×}
[x] ⟼ x^{⊙2} ⊗ (x ∧ (1 − x)) .

Our strategy is as follows: a few preliminary considerations (steps 0.-2.) allow to rewrite the β_{4}-images in a more convenient way, so that the theorem is essentially reduced to Claim (6.1) (in step 3.).

Part I: Reformulations. 0. Note that ϕ(x_{i}) − ϕ(z) is a polynomial in (the variable) z with roots equal to {x_{i}}_{i}, therefore

ϕ(x_{i}) − ϕ(z) = λ ∏_{i}(x_{i} − z)

for some constant λ. Thus, for any fixed l and m,

t − u = ϕ(x_{l}) − ϕ(y_{m}) = λ ∏_{i}(x_{i} − y_{m}) = μ ∏_{j}(x_{l} − y_{j})

for some constants λ and μ (which actually turn out to be equal to ±1 and will be neglected in the following).

1. As a first preliminary step, we express α and 1 − α in terms of the factors x_{i} and y_{j} for each of the arguments α in (4.1). In order to save indices, we put x = x_{ℓ} and y = y_{m} for some fixed ℓ and m. We obtain

\frac{1 − x}{1 − y} = \frac{y^{n−1}X}{x^{n−1}Y}  and  \frac{1 − x^{−1}}{1 − y^{−1}} = \frac{y^{n}X}{x^{n}Y} ,

where we have set X = ∏_{i} x_{i}, Y = ∏_{j} y_{j}. A further decomposition we need is

1 − \frac{1 − x}{1 − y} = \frac{y^{n−1}}{Y} (x − y)  and  1 − \frac{1 − x^{−1}}{1 − y^{−1}} = \frac{y^{n−1}}{xY} (x − y) .

With these preparations, we can write the corresponding images under β_{4} as

β_{4}([\frac{1 − x}{1 − y}]) = (\frac{y^{n−1}X}{x^{n−1}Y} )^{⊗2} ⊗ \frac{y^{n−1}X}{x^{n−1}Y} ∧ \frac{y^{n−1}}{Y} (x − y) (6.1)

and

β_{4}([\frac{1 − x^{−1}}{1 − y^{−1}} ]) = (\frac{y^{n}X}{x^{n}Y} )^{⊗2} ⊗ \frac{y^{n}X}{x^{n}Y} ∧ \frac{y^{n−1}}{xY} (x − y) . (6.2)

These two expressions decompose “naturally” into two parts, one of which contains the factor (x − y) in the last tensor factor. E.g., the part in ⋀^{2} F′^{×} in the first expression (6.1) factors as

\frac{y^{n−1}X}{x^{n−1}Y} ∧ \frac{y^{n−1}}{Y} (x − y) = \frac{X}{x^{n−1}} ∧ \frac{y^{n−1}}{Y} + \frac{y^{n−1}X}{x^{n−1}Y} ∧ (x − y) (6.3)

where the first summand on the right has been reduced using the defining property of ∧.

2. As a second preliminary step, we “switch” to an additive notation—formally, we put ξ_{i} = “ log x_{i}”, η_{j} = “ log y_{j}”, and introduce ζ_{lm} = “ log(x_{l} − y_{m})” as well as the shorthands ξ = ∑_{i} ξ_{i}, η = ∑_{j} η_{j}; then the term (6.1) resp. (6.2) is written, using the decomposition (6.3), as

(6.4) ((n − 1)(η_{m} − ξ_{l}) + ξ − η)^{3} ∧ ζ_{lm}

+ ((n − 1)(η_{m} − ξ_{l}) + ξ − η)^{2} · (ξ − (n − 1)ξ_{l}) ∧ ( − η + (n − 1)η_{m}) resp.

(6.5) (n(η_{m} − ξ_{l}) + ξ − η)^{3} ∧ ζ_{lm}

+ (n(η_{m} − ξ_{l}) + ξ − η)^{2} · (ξ − (n − 1)ξ_{l} + η_{m}) ∧ ( − η + (n − 1)η_{m} − ξ_{l}) .

Introducing the shorthands S = ξ − η and s_{lm} = ξ_{l} − η_{m} , we rewrite the latter two equations as

(S − (n − 1)s_{lm})^{3} ∧ ζ_{lm} + (S − (n − 1)s_{lm})^{2} · (ξ − (n − 1)ξ_{l}) ∧ ( − η + (n − 1)η_{m}) (6.6)

resp.

(S − ns_{lm})^{3} ∧ ζ_{lm} + (S − ns_{lm})^{2} · (ξ − (n − 1)ξ_{l} + η_{m}) ∧ ( − η + (n − 1)η_{m} − ξ_{l}) . (6.7)

Part II: Calculations. 3. So far, we have only reformulated the objects under consideration. Now we can proceed to the two actual calculations involved, the first of which follows easily from inserting the previous expressions: With our conventions from 1. and 2., the β_{4}-image of n^{2}[\frac{1−x_{l}}{1−y_{m}} ] − (n − 1)^{2}[\frac{1−x_{l}^{−1}}{1−y_{m}^{−1}} ] becomes

β_{4}(n^{2}[\frac{1 − x_{l}}{1 − y_{m}} ] − (n − 1)^{2}[\frac{1 − x_{l}^{−1}}{1 − y_{m}^{−1}} ])

= ((2n − 1)S^{3} − 3n(n − 1)S^{2}s_{lm} + n^{2}(n − 1)^{2}s_{lm}^{3}) ∧ ζ_{lm}

−(n^{2}(S − (n − 1)s_{lm})^{2} − (n − 1)^{2}(S − ns_{lm})^{2})( ∑_{i,j}(2 − n)^{δ_{il}+δ_{jm}} ξ_{i} ∧ η_{j}) (6.8)

+(n − 1)^{2}(S − ns_{lm})^{2} · (η_{m} ∧ ξ_{l})
+(n − 1)^{2}(S − ns_{lm})^{2} · (ξ ∧ ξ_{l} + η_{m} ∧ η) .

Here δ_{ij} denotes the usual “Kronecker-δ”.

Summing the latter expression over all l and m, we obtain

Claim 6.1.

β_{4}(n^{2}[\frac{1 − x_{l}}{1 − y_{m}} ] − (n − 1)^{2}[\frac{1 − x_{l}^{−1}}{1 − y_{m}^{−1}} ]) = n(n − 2)(S^{3} ∧ (\frac{1}{n} ∑_{l,m} ζ_{lm}) + S^{2} ∑_{l,m} ξ_{l} ∧ η_{m})

− n^{2}(n − 1)^{2}( ∑_{l,m} s_{lm}^{3} ∧ ζ_{lm} + ∑_{l,m} s_{lm}^{2}(ξ_{l} ∧ η_{m}))

+ ∑_{i} f_{1}(ξ_{i}) + ∑_{j} f_{2}(η_{j})

for some functions f_{k} (k = 1, 2) depending on only one of the variables t and u.

Proof. We will need the following immediately verified identities:

Z := \frac{1}{n} ∑_{i,j} ζ_{ij} = ∑_{i} ζ_{i,m} = ∑_{j} ζ_{lj} , ∀l , ∀m , (6.9)

S = ∑_{i} ξ_{i} − ∑_{j} η_{j} = \frac{1}{n} ∑_{i,j} s_{i,j} = ∑_{l} T_{l,l+k} , ∀k , (6.10)

and the distribution properties (for fixed l, m):

∑_{i,j}(2 − n)^{δ_{il}+δ_{jm}} = ∑_{i}(2 − n)^{δ_{il}} = 1 , (6.11)

as well as some simple consequence of the above

∑_{l,m} s_{lm}(2 − n)^{δ_{il}+δ_{jm}} = S − (n − 1)s_{ij} . (6.12)

[Proof: write s_{lm} = ξ_{l} − η_{m} and use successively the equations in (6.11).]

We observe that, by expanding (S − ns_{lm})^{2} = ((ξ − nξ_{l}) − (η − η_{m}))^{2}, we obtain 3 terms: one purely in the ξ_{i}, a second one purely in the η_{j}, while the third one cancels after summing over l and m:

∑_{l,m}(ξ − nξ_{l})(η − η_{m})(ξ ∧ ξ_{l} + η_{m} ∧ η) = n ∑_{l}(ξ − nξ_{l})(η − η_{m})(ξ ∧ ξ_{l})
−n(ξ − nξ_{l})(η − η_{m})(η_{m} ∧ η) ,

both terms of which are obviously zero.

So it will suffice to concentrate on summing the first three lines in (6.8) over all l and m and identifying the result with the terms given in the first two lines of the claim. We obtain for the sum over all l, m of the first three lines in (6.8):

= (2n − 1)S^{3} ∧ ∑_{l,m} ζ_{lm} − 3n(n − 1)S^{2} ∑_{l,m} s_{lm} ∧ ζ_{lm} + n^{2}(n − 1)^{2} ∑_{l,m} s_{lm}^{3} ∧ ζ_{lm}

− ∑_{l,m} ((2n − 1)S^{2} − 2n(n − 1)Ss_{lm}) ∑_{i,j}(2 − n)^{δ_{il}+δ_{jm}} ξ_{i} ∧ η_{j} (6.13)

+(n − 1)^{2} ∑_{l,m} (S − ns_{lm})^{2}η_{m} ∧ ξ_{l}

(6.14)

which can be simplified using (6.9)-(6.12) as follows: the first two terms combine to

−n(n − 2)S^{3} ∧ Z , (6.15)

the fourth term is divided up into two sums, the first of which is

−(2n − 1)S^{2} ∑_{i,j} ∑_{l,m}(2 − n)^{δ_{il}+δ_{jm}} ξ_{i} ∧ η_{j} = −(2n − 1)S^{2} ∑_{i,j} ξ_{i} ∧ η_{j} , (6.16)

by virtue of (6.11), while the second one is equal to

(6.17) 2n(n − 1)S ∑_{l,m} s_{lm} ∑_{i,j}(2 − n)^{δ_{il}+δ_{jm}} ξ_{i} ∧ η_{j} =

2n(n − 1)S^{2} ∑_{i,j} ξ_{i} ∧ η_{j} − 2n(n − 1)^{2}S ∑_{i,j} s_{ij}(ξ_{i} ∧ η_{j})

where we have used (6.12). Multiplying out the fifth term in (6.13) gives us three further sums, one being quadratic, one linear and one constant in S; the quadratic one combines with (6.16) and the first term in (6.17) to n(n − 2)S^{2} ∑_{i,j} ξ_{i} ∧ η_{j} , the linear one cancels with the second term in (6.17), and the constant one is given by −n^{2}(n − 1)^{2} ∑_{l,m} s_{lm}^{3} ∧ ζ_{lm} . This proves the claim.

4. We finish by observing that the first two terms on the right hand side of Claim 6.1 correspond precisely to the images under β_{4} of n(n − 2)[ ∏ p_{i}/ ∏ q_{j}] and −n^{2}(n − 1)^{2} ∑_{i,j} [p_{i}/q_{j}], respectively, while the last line can be recognized readily—after more careful investigation of the terms “purely in ξ_{i} or η_{j}”— as the image under β_{4} of n^{2}(n − 1)( ∑_{i} [1 − x_{i}^{−1}] − ∑_{i} [1 − y_{i}^{−1}]) .

Acknowledgements: We would like to heartily thank Don Zagier for not only suggesting the problem, but even more for his steady support, encourage- and enlightenment, and constant flow of ideas surrounding this work. It was made possible through the generosity and hospitality of the Max-Planck-Institut für Mathematik in Bonn with its unsurpassed working conditions.

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