\input amstex
%\documentstyle{amsppt}

\documentstyle{amsppt} \nologo \magnification=\magstephalf\TagsOnRight \NoRunningHeads
\def\Q{\Bbb Q} \def\Z{\Bbb Z} \def\N{\Bbb N} \def\C{\Bbb C} \def\R{\Bbb R} \def\O{\text O} 
\def\M{\frak M}  \def\Mk{\M_k} \def\Mkn{\M_k^{\le n}} \def\MRiem{\M^{\text{Riem}}}  \def\dep{\,\text{dep}}
\def\k{\bold k}  \def\A{\Cal A}   \def\v{\varepsilon} \def\z{\zeta} \def\h{\tfrac12} \def\W{S}
\def\MZV{multiple zeta value} \def\endpf{$\quad\blacksquare$\enddemo} \def\qu{\;\overset?\to=\;}

\topmatter\title Multiple Zeta Values\endtitle\author Don Zagier\endauthor\endtopmatter\bigskip

Multiple zeta values are the numbers defined by the convergent series
$$\z(k_1,\ldots,k_n)=\sum_{0<m_1<\ldots<m_n}\frac1{m_1^{k_1}\ldots m_n^{k_n}}\,,\tag 1$$
where $k_1,\ldots,k_n$ are positive integers with $k_n>1$.  Despite this simple-looking definition,
these numbers have deep properties and have appeared in recent years in connection
 with  a surprising diversity of topics, including knot invariants, Galois representations, 
periods of mixed Tate motives, and calculations of integrals associated to Feynman diagrams in
perturbative quantum field theory. 


We will call the number (1) a {\it \MZV} of {\it depth} $n$ and {\it weight} $k$, where $k=k_1+\cdots+k_n$.
Clearly $0<n<k$. There are $\dsize\binom{k-2}{n-1}$ \MZV s of given weight~$k$ and depth $n$ and $2^{k-2}$
altogether of given weight $k$, but the vector space they generate over $\Q$ turns out to have a much smaller
dimension. For example, there are 1024 \MZV s of weight 12, but, according to our high-precision numerical 
calculations, only 12 linearly independent ones.  The main goal of the theory (still very far from being
achieved!) would be to understand all the linear relations over $\Q$ among the \MZV s of a given weight. More precisely,
products of \MZV s can be expressed as a finite integral linear combination of other \MZV s, so that the $\Q$-vector 
space $R$ spanned by all \MZV s forms a $\Q$-algebra, graded by the weight and filtered by the depth, and one would 
like to find its structure as an algebra.

 
The double zeta values ($n=2$) were already studied by Euler, who---as well as his
famous evaluation of the simple zeta values $\z(k)$ as rational multiples of $\pi^k$ when $k$ is even---discovered
empirically, and proved in many cases, that these numbers could be written as rational linear combinations of 
products of two simple zeta values whenever $k_1+k_2$ is odd. Many special identities can be proved. Two of
these, which were conjectured by Hoffman and proved by the author and several other people, are a
``duality formula" saying that  $\z(k_1,\ldots,k_n)=\z(k'_1,\ldots,k'_{n'})$ for a certain involution 
$(k_1,\ldots,k_n)\to(k'_1,\ldots,k'_{n'})$ on index sets, and a ``sum formula" saying that the sum
of the \MZV s of weight $k$ and depth $n$ is equal to $\z(k)$ for any integers $0<n<k$. There are also formulas
describing the value of $\z(\undersetbrace n\to{m,\ldots,m})$ and $\z(\undersetbrace n\to{1,\ldots,1},m)$ for all 
positive integers $n$ and $m$. Yet another identity, which I had conjectured a few years ago and which
was proved only a few weeks ago by D.~Broadhurst, says that
$$ \zeta(\undersetbrace n\to{1,3,\dots,1,3})={2 \pi^{4n}\over (4n+2)!}\,. $$
A simple proof of this using hypergeometric functions is sketched in the lecture. 



The general theory, as already stated, is far from complete. One general result is that any \MZV\ $\z(k_1,\ldots,k_n)$
with $k_1+\ldots+k_n\not\equiv n\pmod2$ is a rational linear combination of products (of the same total weight)
of \MZV s of depth less than $n$ (this includes the two results of Euler for $n=1$ and $n=2$ mentioned above). 
In general, one can show that the number of generators of the ring of \MZV s of weight $k$ and depth $n$
is bounded by $N_n(k-n)$, where $N_n(d)$ ($d>0$) is defined as the dimension of a certain explicitly defined vector 
space $V_n(d)$ of homogeneous polynomials of degree $d$ in $n$ variables. For example, $V_3(d)$ is the space 
of homogeneous polynomials $f(x,y,z)$ of degree $d$ satisfying the two relations
$$ f(x,y,z)+f(x,z,y)+f(z,x,y)=0, \quad\tilde f(x,y,z)+\tilde f(x,z,y)+\tilde f(z,x,y)=0\,,$$
where $\tilde f(x,y,z):=f(x,x+y,x+y+z)$.  The proof of this upper bound comes from the interaction between two sets
of multiplicative relations (the ``double shuffle relations"). For $n=2$ and $n=3$ the dimensions
$N_n(d)$ are nearly completely known:  for $n=2$ (and $d$ even; all $N_n(d)$ for $d$ odd vanish) is equal to
$\lfloor d/6\rfloor$ and for $n=3$ it is $\ge\lfloor(d^2-1)/48\rfloor$, with conjectural equality. The first of 
these two assertions, though it has an elementary proof, is intimately related to modular forms on $PSL(2,\Z)$ and to
the cohomology of this group, and one expects that the theory for general $n$ will be related in a similar way to the 
cohomology of $PSL(n,\Z)$.  The connection between double zeta functions and modular forms can be interpreted
in several different ways and is one of the most intriguing aspects of the whole theory; it is closely related
to the contents of Goncharov's lecture in this conference.


\end

