
2nd semester 2011: Multiple Zeta Values
AMSI Honours Course
A multiple zeta value (MZV) of length $\ell$ and integer weight
$s$ is an $\ell$fold infinite series of the form $$ \zeta(s_1,s_2,\dots,s_\ell)
=\sum_{n_1>n_2>\dots>n_\ell>0}\frac1{n_1^{s_1}n_2^{s_2}\cdots n_\ell^{s_\ell}} $$ where the sum is over $\ell$tuples of
positive integers, and $s_1,s_2,\dots,s_\ell$ are positive integers that add up to $s$ with $s_1>1$. The MZV of length
$\ell=1$ and weight $s$ is just the value $\zeta(s)$ of the Riemann zeta function at $s$, that is, the harmonic series of
exponent $s$. MZVs are also known as multiple harmonic series. They occur in connection with multiple
integrals defining invariants of knots and links, and Drinfeld's work on quantum groups. They also appear in quantum field
theory and throughout combinatorics. MZVs satisfy many striking relations; perhaps the simplest is $$ \zeta(2,1) = \zeta(3)
$$ which goes back to L. Euler (1775). Plan We shall study:  Introduction to multiple
zeta values (MZVs)
 Method of partial fractions
 Algebra of MZVs
 Generalized polylogarithms and generating
functions of MZVs
 Duality theorem. Sum theorem and Ohno's relations
 Quasishuffle products. Derivations

$q$Analogues of MZVs
 Other extensions and open questions
Course Goals
The successful student will emerge from this course
with much enhanced analytic, algebraic and combinatoric skills.
Especially, regarding understanding of:
 proof techniques of identities for MZVs and their various generalisations
 underlying algebraic and combinatorial structures
 computational issues regarding MZVs and polylogarithmic functions
 applications of MZVs
Place and Time
V205 (CARMA, The University of Newcastle)
Time 2:004:00 pm, Wednesday (weekly commencing from Week 2)
Assessment
There will be two graded assignments each counting for
25% of the final mark (Weeks 5 and 9 due), and the final assignment
worth 50% (Week 14 due). Some problems in the assignments will be research problems which will
require from students certain creativity skills including potentials
for experimental mathematics. The students will be expected to produce
their full answers in LaTeX, Maple documents, or similar form.
The assignments should be emailed to us for privacy. Questions can
be discussed with us by email at any time. The due date is always midnight Friday.
Course Notes and Related Material
Resources and Links
 EZFace
 Michael Hoffman's site
contains some basic information about the MZVs.
Hoffman also has a comprehensive
list of references on MZVs and related stuff
 Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr Lisonek,
Special values of multidimensional polylogarithms
Trans. Amer. Math. Soc. 353 (2001), 907–941
 Wadim Zudilin,
Algebraic relations for multiple zeta values
Russian Math. Surveys 58:1 (2003), 1–29
 Jonathan M. Borwein and David M. Bradley,
Thirty Two Goldbach Variations
Int. J. Number Theory 2:1 (2006), 65–103
 David M. Bradley,
Multiple $q$Zeta Values,
Journal of Algebra 283:2 (2005), 752–798
