After recalling the category $\mathcal{MT}(\mathbb{Z})$ of mixed Tate motives over $\mathbb{Z}$ and its relationship with $K$-groups, we discuss the basics of algebraic $K$-theory and Borel's computation of the rational $K$-groups of $\mathbb{Z}$. We then explain how to use Borel's theorem to give a concrete description of the motivic Galois group of $\mathcal{MT}(\mathbb{Z})$ and bound the dimension of the span of multiple zeta values.

This is an expanded note prepared for a STAGE talk, Fall 2014. Our main references are [1],[2],[3], [4] and [5].

TopMixed Tate motives

Let $k$ be a number field with ring of integers $\mathcal{O}$. Last time we introduced a neutral Tannakian category $\mathcal{MT}(k)$ of mixed Tate motives over $k$. Intuitively, it consists of mixed motives over $k$ which are successive extensions of the Tate objects $\mathbb{Q}(n)$. Since the category of mixed motives $\mathcal{MM}(k)$ has yet to be constructed, we had to work around to define $\mathcal{MT}(k)$. People (e.g, Voevodsky) have constructed a triangulated tensor category $\mathcal{DM}(k)$ that behaves like the bounded derived category of the conjectural $\mathcal{MM}(k)$ (even over more general base schemes). We take a triangulated tensor subcategory $\mathcal{DMT}(k)$ generated by the Tate objects. The Beilinson-Soule vanishing conjecture is true for $k$ and it allows one to define a $t$-structure on $\mathcal{DMT}(k)_\mathbb{Q}$ using its natural weight structure. The category of mixed Tate motives $\mathcal{MT}(k)$ is then defined as the heart of this $t$-structure (see [6]). We also introduced a subcategory $\mathcal{MT}(\mathcal{O})\subseteq\mathcal{MT}(k)$ of mixed Tate motives over $\mathcal{O}$. The $\ell$-adic realization of a mixed Tate motive defined over $\mathcal{O}$ is unramified at any place $v\nmid \ell$ of $\mathcal{O}$. These categories can be pictured as follows. 
$$\xymatrix@1@!C=1em{& \mathcal{DM}(k)& \supset &\mathcal{DMT}(k) \ar[d]^{\heartsuit} & &\\ & *+[F--]{\mathcal{MM}(k)} & \supset & \mathcal{MT}(k) & \supset & \mathcal{MT}(\mathcal{O}).}$$
Recall that the subcategory $\mathcal{MT}(\mathcal{O})$ is defined via restricting the extension group $$\Ext^1_{\mathcal{MT}(\mathcal{O})}(\mathbb{Q}(0), \mathbb{Q}(1))=\mathcal{O}^\times \otimes \mathbb{Q} \subseteq k^\times \otimes \mathbb{Q}=\Ext^1_{\mathcal{MT}(k)}(\mathbb{Q}(0),\mathbb{Q}(1)).$$ These can be reinterpreted as the first algebraic $K$-group $$K_1(\mathcal{O})=\mathcal{O}^\times,\quad K_1(k)=k^\times$$ and the restriction can be viewed as a special case of the following theorem, which is a (highly nontrivial) consequence of the construction of $\mathcal{DM}(k)$.

Theorem 1 Let $K_*(k)$ be the algebraic $K$-theory of $k$. Then $$\Ext^1_{\mathcal{MT}(k)}(\mathbb{Q}(0), \mathbb{Q}(n))= K_{2n-1}(k) \otimes \mathbb{Q}$$ and $$\Ext^1_{\mathcal{MT}(\mathcal{O})}(\mathbb{Q}(0), \mathbb{Q}(n))= K_{2n-1}(\mathcal{O}) \otimes \mathbb{Q}.$$ Moreover, all higher $\Ext$-groups vanish.

TopAlgebraic K-theory

We now briefly explain the basics of algebraic $K$-theory (see [7] and [8]) and state Borel's theorem on these rational $K$-groups of $\mathcal{O}$. Algebraic $K$-theory is a sequence of functors $$K_*: \mathbf{Ring} \rightarrow\mathbf{AbGrp},$$ which, roughly speaking, extracts abelian invariants from "linear algebra construction" of $R$.

Definition 1
  • $K_0(R)$ are abelian invariants of "vector spaces" over $R$. The isomorphism classes of finitely generated projective modules over $R$ form a semigroup under the direct sum operation. $K_0(R)$ is defined to be the group completion (aka. Grothendieck group) of this semigroup. The analogy with topological $K$-theory is evident.
  • $K_1(R)$ are abelian invariants of "matrices" over $R$. Let $GL(R)=\cup_{n\ge1} GL_n(R)$, where $GL_n(R)\hookrightarrow GL_{n+1}(R)$ via $g\mapsto \left(\begin{smallmatrix}g & 0\\ 0 & 1\end{smallmatrix}\right)$. Then $K_1(R)$ is defined to be $GL(R)^\mathrm{ab}=GL(R)/[GL(R),GL(R)]$. In fact the commutator $[GL(R),GL(R)]=E(R)$, where $E(R)$ is generated by elementary matrices $e_{ij}(a)$ (1's on the diagonal, $a\in R$ on the $(i,j)$-entry).
  • $K_2(R) $ is defined to be the group homology $H_2(E(R), \mathbb{Z})$, which measures the relations between the elementary matrices $e_{ij}(a)\in E(R)$. Since $E(R)$ is perfect, i.e., $[E(R),E(R)]=E(R)$, we know that $H_2(E(R),\mathbb{Z})$ is isomorphic to the kernel of the universal central extension $\mathrm{St}(R)\rightarrow E(R)$ of $E(R)$. Here $\mathrm{St}(R)$ is known as the Steinberg group of $R$.
Example 1
  • $K_0(k)=\mathbb{Z}$ (= dimension of vector spaces over a field), $K_0(\mathcal{O})=\mathbb{Z} \oplus \Cl(\mathcal{O})$ (each finitely generated projective $\mathcal{O}$-module is isomorphic to a fractional ideal; $I \oplus J=IJ \oplus  \mathcal{O}$).
  • $K_1(k)=k^\times$, $K_1(\mathcal{O})=\mathcal{O}^\times$ (a matrix $A$ over a field or $\mathcal{O}$ can be transformed via elementary matrices into $\diag\{\det A,1,1,\ldots\}$; notice, however, this is not true for a general Dedekind domain $R$).
  • $K_2(k)$ can be identified as Milnor's $K$-group (defined for fields) $$k^\times \otimes_\mathbb{Z} k^\times/\{a \otimes (1-a), a\ne0,1\}.$$ $K_2(\mathcal{O})$ is known to a finite abelian group but harder to compute (it is known that $K_2(\mathbb{Z})=\mathbb{Z}/2 \mathbb{Z}$).
The definition of these lower $K$-groups may seem a bit random at first glance. The good thing is that they fit nicely in long exact sequence (like in usual cohomology theory): $$K_2(R)\rightarrow K_2(R/I)\rightarrow K_1(R, I)\rightarrow K_1(R)\rightarrow K_1(R/I)\rightarrow K_0(R,I)\rightarrow K_0(R)\rightarrow K_0(R/I),$$ where $I\subseteq R$ is an ideal and $K_i(R,I)$ are suitably defined relative $K$-groups. To further extend the long exact sequence to the left, Quillen came up with a uniform definition of all higher $K$-groups rather than constructing them one by one in an ad hoc way. The idea is to realize $K_n(R)$ as the $n$-th homotopy group $\pi_n(X)$ of a certain topological space $X$ (or rather, its homotopy type), which is constructed from linear algebra over $R$ so that the $K_n(R)$ ($n\le 2$) matches up with the previous definitions. The above long exact then comes for free from the long exact homotopy sequence associated to a pair of spaces. Quillen gave two constructions:
  • $X=BGL(R)^+\times K_0(R)$, where $BGL(R)^+$ is the $+$-construction applied to the classifying space $BGL(R)$ of $GL(R)$. The $+$-construction of a topological space modifies its fundamental group but does not change its homology;
  • $X=\Omega B \mathcal{Q}R$. Here $\mathcal{Q}R$ is a category whose objects are finitely generated projective $R$-modules and $\Hom(P_1,P_2)$ is the set of equivalence classes of diagrams $$P_1\twoheadleftarrow Q \hookrightarrow P_2$$ such that $\ker(Q\twoheadrightarrow P_1)$ and $\coker(Q\hookrightarrow P_2)$ are also finitely generated $R$-modules. $\Omega B\mathcal{Q}R$ is the loop space of the classifying space of the category $\mathcal{Q}R$: so $K_n(R)=\pi_{n+1}(B\mathcal{Q}R)$.

These two constructions look very different on the surface, but they turn out to produce the same $K$-groups.

Example 2 For a finite field $\mathbb{F}_q$, Quillen computed that $K_{2n}(\mathbb{F}_q)=0$ and $K_{2n-1}(\mathbb{F}_q)=\mathbb{Z}/(q^n-1)$ for $n\ge1$.

When $R=\mathcal{O}$ is the ring of integers of a number field $k$, Quillen proved that $K_n(\mathcal{O})$ is always finitely generated. A celebrated theorem of Borel further computed the rank of $K_n(\mathcal{O})$.

Theorem 2 (Borel) Suppose $r_1$ and $r_2$ is the number of real and complex places of $k$. For $n\ge 1$, $$\dim K_n(\mathcal{O})\otimes \mathbb{Q}=
    0, & n \text{ is even}, \\
    r_1+r_2-1, & n=1, \\
    r_2, & n=3,7, 11\ldots,\\
    r_1+r_2, & n=5, 9, 13,\ldots.
Remark 1 For $n=1$, Borel's theorem recovers Dirichlet's unit theorem.
Remark 2 Higher $K$-groups are generally very difficult to compute. A useful consequence of Quillen's $Q$-construction is the localization theorem for computing $K$-groups. For example, for any Dedekind domain $R$ with fraction field $F$, there is a long exact sequence $$\cdots\rightarrow K_n(R)\rightarrow K_n(F)\rightarrow \bigoplus_{\mathfrak{p}} K_{n-1}(R/\mathfrak{p})\rightarrow K_{n-1}(R)\rightarrow\cdots,$$ where $\mathfrak{p}$ runs over all maximal ideals of $R$. For $R=\mathcal{O}$, more is true: Soule proved that the map $K_n(\mathcal{O})\rightarrow K_n(F)$ is injective, therefore by Example 2for each $n\ge2$, we have $$K_{2n-1}(\mathcal{O})= K_{2n-1}(k)$$ and for each $n\ge1$ an exact sequence $$0\rightarrow K_{2n}(\mathcal{O})\rightarrow K_{2n}(k)\rightarrow \bigoplus_{\mathfrak{p}} K_{2n-1}(\mathcal{O}/\mathfrak{p})\rightarrow0.$$ In particular, by Borel's theorem, $K_{2n}(k)$ is an infinite torsion group.
Remark 3 Quillen's construction can be extended to any scheme $X$. The Chern classes of algebraic $K$-theory $c^{q,p}: K_{2q-p}(X)\rightarrow H^p_\mathrm{mot}(X, \mathbb{Z}(q))$ induce an isomorphism $$K_{2q-p}(X)^{(q)}\cong H^p_\mathrm{mot}(X, \mathbb{Z}(q)) \otimes \mathbb{Q}:=\Hom_{\mathcal{DM}(k)}(\mathbb{Q}(0), \mathbb{Q}_X(q)[p]).$$ Here $K_*(\cdot)^{(q)}$ is the weight $q$ eigenspace of the Adams operators acting on $K_*(\cdot)\otimes \mathbb{Q}$ and $\mathbb{Q}_X(q)$ is an object in $\mathcal{DM}(k)$. Theorem 1 is a special case when $X=\Spec k$, $p=1$ and $q=n$ since $$K_{2q-1}(k)^{(q)}=K_{2q-1}(k) \otimes \mathbb{Q}$$ by a result of Soule. Also, for $n\ge2$, the extension group $\Ext^1(\mathbb{Q}(0), \mathbb{Q}(n))$ doesn't change when restricting to $\mathcal{MT}(\mathcal{O})$. Indeed we have $K_{2n-1}(k)=K_{2n-1}(\mathcal{O})$ when $n\ge2$ as in the previous remark.
Remark 4 For $n\ge1$, the Dedekind zeta function $\zeta_k(s)$ has order of vanishing equal to $\dim K_{2n-1}(\mathcal{O}) \otimes \mathbb{Q}$ at $s=1-n$. More generally, Beilinson's first conjecture predicts the relation between order of vanishing of motivic $L$-function and motivic cohomology groups when $2n\le i$: $$\ord_{s=n} L(H^i(X),s)=\dim H^{i+1}_\mathrm{mot}(X, \mathbb{Q}(i+1-n)).$$ From this point of view, Borel's theorem verifies the special case $X=\Spec k$, $i=0$.
Remark 5 The torsion part of $K_n(\mathcal{O})$ is harder to compute. People knew the case when $n$ is odd. People also knew the order of $K_{4k+2}(\mathbb{Z})$ and conjectured that $K_{4k+2}(\mathbb{Z})$ should be a cyclic group. The order $|K_{4k+2}(\mathbb{Z})|$ is related to the $k$-th Bernoulli number: no surprise they should relate to the Riemann zeta function as well (Lichtenbaum's conjecture)! People also conjectured that $K_{4n}(\mathbb{Z})=0$ — this is in fact equivalent to Vandiver's conjecture on class groups of cyclotomic fields and seems to be extremely difficult.

TopMotivic Galois group of $\mathcal{MT(\mathcal{O})}$

Recall that the de Rham realization functor $$\omega: \mathcal{MT}(\mathcal{O})\rightarrow \mathbf{Vect}_\mathbb{Q},\quad \omega(M)=\bigoplus_{n\in \mathbb{Z}} \omega_n(M)$$ is a fiber functor (i.e., a faithful $\mathbb{Q}$-linear exact tensor functor). Here $$\omega_n(M)=\Hom(\mathbb{Q}(n), W_{-2n}(M)/W_{-2(n+1)}(M))$$ is multiplicity space of the weight $-2n$ piece of $M$. This fiber functor $\omega$ makes $\mathcal{MT}(\mathcal{O})$ is neutral Tannakian category. We denote by $G=G_\omega(\mathcal{MT}(\mathcal{O}))$ the corresponding Tannakian fundamental group. Then $G$ is a pro-algebraic group defined over $\mathbb{Q}$ and $\mathcal{MT}(\mathcal{O})$ is equivalent to $\mathbf{Rep}(G)$, the category of finite dimensional $\mathbb{Q}$-representations of $G$. Naturally $G$ is called the motivic Galois group of $\mathcal{MT}(\mathcal{O})$. By definition we have $$H^i(G, \mathbb{Q}(n))=\Ext_{\mathcal{MT}(\mathcal{O})}^i(\mathbb{Q}(0), \mathbb{Q}(n)).$$ Our next goal is to give a concrete description of $G$ using this relation between the group cohomology of $G$ and the extension groups computed by Borel's theorem.

For any object $M\in \mathcal{MT}(\mathcal{O})$, we denote by $\langle M\rangle$ the full Tannakian subcategory generated by $M$ (whose objects are subquotients of $M^{\otimes p} \otimes (M^\vee)^{\otimes q}$). If $\langle M_1\rangle\subseteq \langle M_2\rangle$, then we have a surjection $G_\omega\langle M_2\rangle\rightarrow G_\omega\langle M_1\rangle$, which induces an isomorphism $$G=\varprojlim_M G_\omega\langle M\rangle.$$ The "smallest" nontrivial Tannakian subcategory $\langle \mathbb{Q}(1)\rangle\subseteq \mathcal{MT}(\mathcal{O})$ is nothing but the category of pure Tate motives over $\mathcal{O}$. Notice that $G_\omega\langle \mathbb{Q}(1)\rangle$ is determined by its action on the 1-dimensional vector space $\omega(\mathbb{Q}(1))$, hence is $\mathbb{G}_m$. This induces a surjection $G\rightarrow \mathbb{G}_m$ and we denote its kernel by $U$. Since $U$ acts trivially on each graded piece of $\omega(M)$, $U$ is pro-unipotent. Moreover, the exact sequence $$1\rightarrow U\rightarrow G\rightarrow \mathbb{G}_m\rightarrow1$$ in fact splits: a splitting $\mathbb{G}_m\rightarrow G$ is given by $\lambda\mapsto$ multiplication by $\lambda^n$ on $\omega_n(M)$. This identifies $G$ as the semi-direct product $$G=U \rtimes \mathbb{G}_m.$$ It remains to compute $U$:

Theorem 3 $U$ is a free pro-unipotent group generated by $\dim K_{2n-1}(\mathcal{O}) \otimes \mathbb{Q}$ elements in degree $n$. In particular, when $\mathcal{O}=\mathbb{Z}$, it follows from Borel's theorem that $U$ is a free pro-unipotent group with one generator in each odd degree $n\ge3$.

Let us review a bit background on pro-unipotent completions (see [9]). For any abstract group $\Gamma$, its group algebra $\mathbb{Q}[\Gamma]$ is naturally a Hopf algebra under $$\Delta: g \mapsto g \otimes g,\quad s: g\mapsto g^{-1}, \quad \varepsilon: g\mapsto 1.$$ Moreover, the group $G$ can be recovered as the group-like elements of the Hopf algebra $R=\mathbb{Q}[\Gamma]$, $$G=\mathcal{G}(R)=\{x\in R^\times: \Delta x=x \otimes x\}.$$ These two functors form an adjoint pair $$\mathbb{Q}[\cdot]\{\text{Abstract groups}\}\rightleftarrows\{\text{Hopf algebras}\}: \mathcal{G}.$$ Notice that the Hopf algebra $\mathbb{Q}[\Gamma]$ is cocommutative not necessarily commutative (it is commutative if and only if $\Gamma$ is abelian), hence does not correspond to the coordinate ring of an algebraic group. To obtain a commutative Hopf algebra, the natural idea is to take dual. To ensure taking dual is a reasonable operation, the topology on such Hopf algebras should not be too far away from finite dimensional vector spaces.

Definition 2 A topological vector space $V$ is linearly compact if it is homeomorphic to $\varprojlim V/V_i$, where $V/V_i$ are discrete and finite dimensional. If $V$ is linearly compact and $V^*$ is its topological dual, then the linear dual $(V^*)^\vee=V$.

For a Hopf algebra $R$ with argumentation ideal $I$, the $I$-adic completion $\hat R=\varprojlim R/I^k$ is is linearly compact if and only if $I/I^2$ is finite dimensional. Restricting to such a subcategory we obtain a functor $${\{\text{Abstract group }\Gamma,\atop \Gamma^\mathrm{ab} \otimes_\mathbb{Z} \mathbb{Q}\text{ finite dimensional}\}}\rightarrow \{\text{Pro-unipotent algebraic groups}\},$$ given by $\Gamma\mapsto \Spec \widehat{\mathbb{Q}[\Gamma]}^*$. Moreover, for a unipotent algebraic group $G=\Spec R$, $G(\mathbb{Q})=\mathcal{G}(R^\vee)$ gives an adjoint functor (but not an equivalence).

Example 3 Suppose $\Gamma$ is a free abstract group on two generators $x_0,x_1$.
  • $\mathbb{Q}[\Gamma]=\mathbb{Q}\langle x_0,x_1\rangle$ is the non-commutative polynomial ring in $x_0,x_1$.
  • $\widehat{\mathbb{Q}[\Gamma]}=\mathbb{Q}\langle\langle y_0,y_1\rangle\rangle$ is the non-commutative power series ring in $y_0=x_0-1,y_1=x_1-1$. As a graded vector space, $\widehat{\mathbb{Q}[\Gamma]}$ is isomorphic to the $\prod_{k\ge0} V^{\otimes k}$, where $V$ is a 2-dimensional vector space with basis $\{y_0,y_1\}$.
  • $G=\Spec \widehat{\mathbb{Q}[\Gamma]}^*$ is the free pro-unipotent group in two generators. As a graded vector space, $\Spec \widehat{\mathbb{Q}[\Gamma]}^*=\bigoplus_{k\ge0}(V^\vee)^{\otimes k}$. The algebra structure is given by the shuffle product and the coalgebra structure is given by de-concatenation.
  • $G(\mathbb{Q})$ consists of group-like power series in $(\widehat{\mathbb{Q}[\Gamma]}^*)^\vee=\widehat{\mathbb{Q}[\Gamma]}=\mathbb{Q}\langle\langle y_0,y_1\rangle\rangle$.
Remark 6 Analogous to free groups, free pro-unipotent groups can be characterized by the lifting property: if $1\rightarrow G_1\rightarrow G_2\rightarrow G_3\rightarrow 1$ is an exact sequence of pro-unipotent groups, for a free pro-unipotent group $F$, any homomorphism $F\rightarrow G_3$ lifts to a homomorphism $F\rightarrow G_2$. Also analogous to usual group cohomology, for a pro-unipotent group $G$, $H^1(G, \mathbb{Q})$ (resp., $H^2(G, \mathbb{Q})$) has an interpretation of generators (resp., relations). In particular, if $H^2(G, \mathbb{Q})=0$, then $G$ is a free pro-unipotent group, with generators given by a basis of $H^1(G,\mathbb{Q})$ (see [10]).

Now let us come back to our original situation $G=U \rtimes \mathbb{G}_m$, where $G$ is the motivic Galois group of $\mathcal{MT}(\mathcal{O})$. The cohomology $H^i(U,\mathbb{Q})$ can be computed as follows. Theorem 3 then follows from it in view of the previous Remark 6 and Theorem 1.

Theorem 4 We have an isomorphism of $\mathbb{G}_m$-representations $$H^i(U,\mathbb{Q})=\bigoplus_n \Ext^i_{\mathcal{MT}(\mathcal{O})}(\mathbb{Q}(0), \mathbb{Q}(n)) \otimes \mathbb{Q}(-n).$$
Proof For any $G$-representation $V$, the Hochschild-Serre spectral sequence implies that $$E_2^{p,q}=H^p(\mathbb{G}_m, H^q(U,V))\Rightarrow H^{p+q}(G, V).$$ Since $\mathbb{G}_m$ is reductive, we have $H^p(\mathbb{G}_m, -)=0$ for $p\ge1$. Therefore, $$H^0(\mathbb{G}_m, H^i(U,V))=H^i(G,V).$$ Now take $V=\mathbb{Q}(n)$, we obtain that $$H^i(U, \mathbb{Q}(n))^{\mathbb{G}_m}=H^1(G, \mathbb{Q}(n))=\Ext^i_{\mathcal{MT}(\mathcal{O})}(\mathbb{Q}(0), \mathbb{Q}(n)).$$ Since $H^i(U,\mathbb{Q}(n))=H^i(U,\mathbb{Q}) \otimes \mathbb{Q}(n)$, we obtain that $$H^i(U,\mathbb{Q})=\bigoplus_n \Ext^i_{\mathcal{MT}(\mathcal{O})}(\mathbb{Q}(0), \mathbb{Q}(n)) \otimes \mathbb{Q}(-n),$$ as desired.

TopMultiple zeta values

Finally, we will use the concrete description of $U$ in Theorem 3 to give an explicit upper bound for the span of multiple zeta values. Recall that for $n_1,\ldots n_r\ge1$ such that $n_r\ge2$, the multiple zeta values is defined to be $$\zeta(n_1,\ldots,n_r)=\sum_{1\le k_1< \cdots< k_r} \frac{1}{k_1^{n_1}\cdots k_r^{n_r}},$$ where $r$ is called its depth and $N$ is called its weight. The depth $r=1$ case recovers the classical zeta values $\zeta(n)$, $n\ge2$. These are transcendental real numbers satisfying a lot of mysterious relations (see [11]). We are interested in finding the dimension of the $\mathbb{Q}$-linear span $$\mathrm{MZV}_N=\mathbb{Q}\{ \zeta(n_1,\ldots, n_r): n_1+\cdots+n_r=N\}$$ of multiple zeta values of weight $N$.

Example 4 The multiple zeta value can be viewed as an iterated integral as follows. Let $w=w_1\cdots w_n$ be a word in 0 and 1 starting with 1 and ending with 0. Let $$I(w)=\int_{0\le t_1\le t_2\cdots \le t_n\le1}\frac{dt_1}{t_1-w_1}\cdots \frac{d t_n}{t_n-w_n}.$$ Then it is easy to see that $$\zeta(n_1,\ldots,n_r)=(-1)^rI(10^{n_1-1}10^{n_2-1}\ldots 10^{n_r-1}).$$ From this integral formula one can deduce that the product of two multiple zeta values with words $w_1$ and $w_2$ can be written as an $\mathbb{Z}$-linear combinations of other multiple zeta values (by shuffling $w_1$ and $w_2$). For example, $\zeta(2)=-I(10)$, $\zeta(2,2)=I(1010)$, $\zeta(1,3)=I(1100)$. The shuffle product of $10$ with $10$ is $2\cdot 1010+4\cdot 1100$ which implies the relation $$\zeta(2)^2=2\cdot\zeta(2,2)+4\cdot\zeta(1,3).$$ These are known as shuffle relations.
Example 5 From the original definition one can also write the product of two multiple zeta values as an $\mathbb{Z}$-linear combination by reordering the summation. For example, $$\zeta(2)^2=2\cdot\zeta(2,2)+\zeta(4).$$ These relations are known as stuffle relations.
Example 6 Another more striking relation is given by the sum theorem (proved by Granville and Zagier independently). It says that the sum of multiple zeta values with the same weight $N$ and depth $r$ are the same when varying the depth $r$. For example, $$\zeta(4)=\zeta(1,3)+\zeta(2,2)=\zeta(1,1,2).$$

The following remarkable conjecture of Zagier predicts that though there are $2^{N-2}$ multiple zeta values with weight $N$, the $\mathbb{Q}$-span of them is much smaller.

Conjecture 1 (Zagier) Define $d_0=1$, $d_1=0$, $d_2=1$ and $d_k=d_{k-2}+d_{k-3}$ for $k\ge3$. Then $\dim \mathrm{MZV}_N= d_N$.
Example 7 The shuffle relation, the stuffle relation and the sum relation together implies that $$\zeta(4)=\zeta(1,1,2)=4\cdot\zeta(1,3)=\frac{4}{3}\zeta(2,2)=\frac{2}{5}\zeta(2)^2.$$ In particular, $\dim \mathrm{MVZ}_4=1$ as predicted (these are all rational multiples of $\pi^4$!) Zagier's conjecture can be checked for small values of $N$ (this is done at least for $N\le12$), which probably serves as a good reason for making such a conjecture.
Example 8 There are $2^{28}=268435456$ multiple zeta values of weight $N=30$, but $\dim \mathrm{MZV}_{30}$ is conjecturally only $d_{30}=1897$.

Using the concrete description of pro-unipotent group $U$, we are now able to prove the following upper bound.

Theorem 5 $\dim \mathrm{MZV}_N\le d_N$.
Proof (Idea of the proof) Mimicking the iterated integrals defining the multiple zeta values, one can define motivic iterated integral using the theory of motivic fundamental group of $\mathbb{P}^1-\{0,1,\infty\}$, and then define a graded algebra $\mathcal{H}$ consisting of motivic multiple zeta values $\zeta^\mathrm{mot}(n_1,\ldots,n_r)$ whose image under a period map are the classical multiple zeta values $\zeta(n_1,\ldots,n_r)$.

More precisely, let $_0\Pi_1=\pi_1(\mathbb{P}^1-\{0,1,\infty\}, \vec 1_0,-\vec 1_1)$ be the motivic torsor of paths from 0 to 1. Evaluating a regular function on ${}_0\Pi_1$ at the straight path $\mathrm{dch}$ (droit chemin in French) from 0 to 1 gives a homomorphism $$\mathrm{dch}:\mathcal{O}(_0\Pi_1)\rightarrow \mathbb{R}.$$ Concretely, $\mathrm{dch}\in {}_0\Pi_1(\mathbb{R})$ is a group-like power series in two letters 0 and 1 (see Example 3) whose coefficient before a word $w$ is given by the iterated integral $I(w)$. Recall that $\mathcal{O}({}_0\Pi_1)$ is an ind-object in $\mathcal{MT}(\mathbb{Z})$ and thus $U$ acts on it. Then $\mathcal{H}$ is defined to be the quotient of $\mathcal{O}({}_0\Pi_1)$ by the "motivic relations" between multiple zeta values: namely the quotient $\mathcal{O}({}_0\Pi_1)/J$ by the largest ideal $J\subseteq \ker (\mathrm{dch})$ which is stable under the action of $U$.

By Theorem 3, we have $$\mathcal{O}(U)=\mathbb{Q}\langle f_3,f_5,f_7,\ldots\rangle$$ as a graded vector space (where $f_i$ has degree $i$). It turns out that one can find a (non-canonical) injective $\mathcal{O}(U)$-comodule morphism $$ \mathcal{H}\hookrightarrow \mathcal{H}^{\mathcal{MT}_+}:=\mathcal{O}(U) \otimes_\mathbb{Q} \mathbb{Q}[f_2]$$ which maps $\zeta^\mathrm{mot}(2)$ to $f_2$ and $\zeta^\mathrm{mot}(2n+1)$ to $f_{2n+1}$. Clearly an upper bound on the dimension of the weight $N$ piece $\mathcal{H}^{\mathcal{MT}_+}_N$ of $\mathcal{H}^{\mathcal{MT}_+}$ gives an upper bound on $\dim \mathcal{H}_N$, hence on $\dim\mathrm{MVZ}_N$.

Now the point is that we can compute $\dim \mathcal{H}^{\mathcal{MT}_+}_N$ explicitly! It is the coefficients of $t^N$ in the generating series $$\left(\sum_{k\ge0}(t^3+t^5+t^7\cdots )^k\right)\left(\sum_{k\ge0} (t^2)^k\right),$$ which is equal to $$\frac{1}{1-t^3-t^5-\cdots}\cdot \frac{1}{1-t^2}=\frac{1}{1-\frac{t^3}{1-t^2}}\cdot\frac{1}{1-t^2}=\frac{1}{1-t^2-t^3}.$$ This last generating series is nothing but $\sum_{N\ge0} d_Nt^N$!


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