Project IV (MATH4072) 2016-2017

The algebra of multiple zeta values

Herbert Gangl


Euler showed that the sum of the inverse squares over all natural numbers equals π2/6 and more generally, for any even integer n>1, that the corresponding sum of the inverses of n-th powers is a rational multiple of πn. Such sums, called zeta values, play an important role in number theory. In recent years, iterated sums of a similar kind, the so-called multiple zeta values (MZVs), have appeared in many different contexts, ranging from number theory, algebraic geometry or knot theory to Feynman integrals in quantum field theory.

There is rather little known about individual evaluations of these MZVs; by contrast, they satisfy a wealth of interesting relations like two types of multiplication as well as a so-called co-multiplication, and moreover one knows many combinatorial identities like sum formulas, "derivation formulas" or relations coming from a certain difference equation, the "Newton series". It is one of the basic problems in the theory to exhibit a complete set of such relations, at least two candidates of which have been proposed (and recently related to each other).

An important feature which guarantees a lot of structure is that each such multiple zeta value can be written both as an iterated sum and as an integral of a very specific kind (one iteratedly integrates rather simple rational functions).

Some of the rich structure behind these numbers can be encoded in terms of, e.g., shuffle algebras, Lie algebras as well as Hopf algebras, the latter of which are delicate objects in which one can compose (i.e. multiply) and decompose (i.e. "co-multiply"). The algebras attached to MZVs have a strong combinatorial flavour.

Moreover, there is a surprising connection to other--a priori completely unrelated--number theoretic objects, so-called modular forms, which give rise, via their "period polynomials", to special relations among multiple zeta values. These special relations in turn relate modular forms to conjectures on the dimension of rational vector spaces spanned by the MZVs.

Intriguingly, multiple zeta values very often appear as coefficients in expansions of integrals attached to Feynman graphs which arise, e.g., in quantum field theory calculations. This hints at an interplay between number theory and physics, and in recent years such parallels and connections have indeed been discovered and developed. Moreover, the last few years witnessed several breakthrough results in the field, in particular by Furusho and Brown, some of which should be accessible for a highly dedicated student.

Possible directions

As there are many different directions and levels on which multiple zeta values can be studied, from the very concrete to rather abstract, this project can easily be taylored to the student's background. Depending on the topic chosen the project could involve "experimental" computer work, in particular the algebra/number theory package GP/PARI which has many MZV routines built in, but it could also involve theoretical investigation or a combination of the two.


Familiarity with analysis of sums and series, as well as partial fraction expansions, is recommended; for a study of the underlying algebraic structures, Algebra would be needed, and additional accompanying Number Theory would certainly be very helpful.


Original papers: there are the following often cited articles

  • M. E. Hoffman, `Multiple harmonic series,' Pacific J. Math. 152 (1992), 275-290.
  • D. Zagier, `Values of zeta functions and their applications,' in First European Congress of Mathematics (Paris, 1992), Vol. II, A. Joseph et. al. (eds.), Birkhäuser, Basel, 1994, pp. 497-512.

    A rather comprehensive web page of references for all kinds of results on multiple zeta values, including a reference to Euler's original work (Ref. A.2 there), is

  • M. E. Hoffman: References on multiple zeta values.

    Among the papers listed there is a nice survey of certain aspects, given by Zudilin in

  • W. Zudilin, `Algebraic relations for multiple zeta values' (Russian), Uspekhi Mat. Nauk 58 (2003), 3-32; English translation in Russian Math. Surveys 58 (2003), 1-29; preprint, no.33 of Section C on said web page,

    and there was a lecture series on the subject given by Waldschmidt

  • M. Waldschmidt, `Lectures on multiple zeta values'

    A further recommended article (alas, only in French) is

  • P. Cartier, `Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents,' Astérisque 282 (2002), 137-173 (Sém. Bourbaki no. 885).

    The shuffle relations are also well explained e.g. in Hoffman, `The algebra of multiple harmonic series,' J. Algebra 194 (1997), 477-495.

    As a starting point for rather elementary reading, one can admire the many ways to obtain Euler's first surprising identity ζ(2,1)= ζ(3), a collection of proofs is presented in

  • J.Borwein, D. Bradley: Thirty-two Goldbach variations

    and many other combinatorial articles can be found, e.g.

  • J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, `Combinatorial aspects of multiple zeta values,' Electronic J. Combinatorics 5 (1998), R38.

    Connections of multiple zeta values to modular forms, very different and highly important number theoretical objects, can be found in

  • H. Gangl, M. Kaneko, D. Zagier, `Double zeta values and modular forms,' in Automorphic forms and zeta functions, S. Böcherer et. al. (eds.), World Scientific, Hackensack, NJ, 2006, pp. 71-106.

    Some of the probably earliest occurrences of multiple zeta values in Feynman graph calculations are in

  • J. M. Borwein and R. Girgensohn, `Evaluation of triple Euler sums,' with appendix `Euler sums in quantum field theory' by D. J. Broadhurst, Electronic J. Combinatorics 3 (1996), R23.
  • D. J. Broadhurst and D. Kreimer, `Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops,' Physics Lett. B 393 (1997), 403-412.

    A good basis for performing numerical experiments can be found in the following sources:

  • R. E. Crandall, `Fast evaluation of multiple zeta sums,' Math. Comp. 67 (1998), 1163-1172.
  • R. E. Crandall and J. P. Buhler, `On the evaluation of Euler sums,' Experiment. Math. 3 (1994), 275-285.
  • email: Herbert Gangl