Multiple zeta values and their relations
Created by Henrik Bachmann (Nagoya University)

This poster can be downloaded at www.henrikbachmann.com

Notation
For k1, . . . , kr−1 ≥ 1, kr ≥ 2 the multiple zeta values are defined by11
ζ(k1, . . . , kr ) =k1
kr
m.
.
.
mr
10<m1<···<mrX4
ζ(4) = 3ζ(2, 2) = 4ζ(1, 3) = ζ(1, 1, 2)

Motivic relations

Z: Q-algebra of multiple zeta values
Zk : space of multiple zeta values of weight k = k1 + · · · + kr2

Algebraic setup:5ζ(5) = 6ζ(3, 2) + 4ζ(1, 2, 2)

Associator relations
ΦKZ(e0, e1)

H = Qhe0, e1i , H0 = Q + e1He0 , H1 = Q + e1H
ζ : H0 −→ Zw = e1ek01−1 · · · e1ek0r −1 7−→ ζ(w) := ζ(k1, . . . , kr )2π
ζ(2) = 625

Integral-Series identity
ζ(µ(k, l)) = ζ(k  l?)44n12π
ζ({1, 3}n) = 4n ζ({4}n) = (4n+2)!

24

Confluence relations
L(ICF) = 03

ζ(k) = ζ(1,.
.
.
,1,2)
| {z }

4π22
ζ(4) = 5 ζ(2) = 90

 or ∗

k−2

Regularized double shuffle relations
01w
∈
H,v∈
H5ζ(reg(w v − w ∗ v)) = 0

Euler’s relations
8
ζ(2k) ∈ ζ(2)k Q = π 2k Q−(2πi)2k B2k
ζ(2k) = 2(2k)!

Dualityζ(w) = ζ(τ (w))

7



26

ζ(2)ζ(3) = ζ(3, 2) + 3ζ(2, 3) + 6ζ(1, 4)

ζ(3, 2) = ζ(2, 1, 2)
6

Shuffle product10 ζ(w)ζ(v) = ζ(w v)

Hoffman’s relations
21 ζ(w e − w ∗ e ) = 0
11Linear part ofFinite double shuffle
9 ζ(w v − w ∗ v) = 0

w, v ∈ H0

Kawashima’s relations

Harmonic product11 ζ(w)ζ(v) = ζ(w ∗ v)

1327

23

Rooted tree maps
14
∈ End(H1)

Kawashima’s relations

Quasi-derivation relations
16(c)ζ(∂n (w)) = 0

12

Ohno’s relations

ζ(2)ζ(3) = ζ(2, 3) + ζ(3, 2) + ζ(5)30

Ohno-Zagier relations
15

X

28

Derivation relations
17ζ(∂n(w)) = 0

ζ(k) ∈ Q[ζ(l) | l ≤ k]29

wt(k)=kdep(k)=rht(k)=s

20 Weighted sum formula
31

19 Cyclic sum formula
Linear relations

18 Restricted sum formula
Algebraic relations

ζ(3) = ζ(1, 2)

Conjecturally imply all5197
28ζ(3, 9) + 150ζ(5, 7) + 168ζ(7, 5) = 691 ζ(12)relations among MZV

Sum formula
A

22

B

ζ(k1, . . . , kr ) = ζ(k)k1+···+kr =kk1,...,kr−1≥1,kr ≥2

A implies B(after a possible use ofX

 or ∗ )Conjectured dimensions and number of linearly independent relations
32

Conjecture (Zagier)weight k0 1 2 3 4 5 6 7 8910 11121314# of generators 1 0 1 2 4 8 16 32 64 128 256 512 1024 2048 4096

The generating series of the dimension of Zk is given byX1kdimQ Zk X =.
231−X −X?

# of relations = 0 0 0 1 3 6 14 29 60 123 249 503 1012 2032 4075?

dimQ Zk =k≥0In particular for k ≥ 3 we have dimQ Zk = dimQ Zk−2 + dimQ Zk−3.

1 0 1 1 1 2234579121621

Number of generators, conjectured number of linearly independent relations and conjectured dimension of Zk

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