Priority Research Centre for Computer-Assisted
Research Mathematics and its Applications

2nd semester 2011: Multiple Zeta Values

Jonathan M. Borwein and Wadim Zudilin

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
• Quasi-shuffle 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

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

• Introduction
• Lecture 1 (August 3)
• Lecture 2 (August 10)
• Lecture 3 (August 17)
• Lecture 4 (August 24)
• Assignment One: due Friday August 26 comprises Exercises 1.1 (p.2), 3.1 (p.8), 4.1 (p.13), 5.3 and 5.4 (p.18)
• Lecture 5 (August 31)
• Lecture 6 (Power Point) and Parametric Euler sums (September 7)
• Lecture 7 (September 14)
• Lecture 8 (September 21)
• Assignment Two: due Friday September 23 comprises Exercises 9.1 (p.33), 10.2 (p.34), 10.5 (p.34), and Jon's question from Week 6 (see p.15 in Lecture 6): Identify 0.438017879485942412114...
• Lecture 9 (October 12)
• Lecture 10 (October 19)
• Final Assignment: due Friday November 11 comprises Exercises 2.1(a) (p.6), 11.1 (p.37), 15.1 (p.52), 16.1 (p.56)

Resources and Links

• EZ-Face
• 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
  

• Jon's Mathematics Portal
• Experimental Mathematics website
  • and Math Drudge blog