Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values

The following sections are included:

• Dedication
• Contents
• Preface
• Introduction

The multiple zeta functions are nested generalizations of the Riemann zeta function. In this chapter we start by summarizing briefly the most important results of the Riemann zeta function. Then we bring in one of our major players in this book — the multiple zeta function — and outline its main properties and problems. Some results will be generalized to higher levels in Chap. 13, but often with more advanced and different proofs. So, we still provide complete proofs of these results in this chapter.

In this chapter we consider single- and multi-variable generalizations of the natural logarithm, called the polylogarithms (polylogs) and the multiple polylogarithms (multiple polylogs or MPLs), respectively.

Most of the on-going studies on multiple zeta functions concentrate on the algebraic aspect of the multiple zeta values (MZVs for abbreviation), which are the convergent special values of the multiple zeta functions at positive integers. Sometimes MZVs are also called multi-zeta values, or poly-zeta values, or Euler–Zagier sums.

A crucial observation by Deligne is that one can attach tangential base point on ℙ \ {0, 1, ∞} to study the torsors of paths and the motivic fundamental group. Let 1₀ and (−1)₁ be the tangential base point at 0 (resp. at 1) pointing to the positive (resp. negative) direction. Then the de Rham version of the Betty path from 1₀ going straight to (−1)₁ gives rise to the Drinfeld associator whose coefficients are all expressible by the MZVs.

There are plenty of known relations among the MZVs most of which have been proved rigorously in recent years with a variety of techniques. In Sec. 3.3 we have seen that the (regularized) DBSFs give rise to many linear relations over ℚ and conjecturally these imply all the other linear relations. In this chapter we first collect a few special families of these relations and then present some conjectural ones at the end. We will often resort to the properties of Bernoulli numbers thanks to Euler's famous evaluation in Eq. (1.2).

In the preceding two chapters, we have considered many interesting properties of the MZVs. However, one unpleasant feature causes a lot of trouble, namely, admissible MZVs must start with an argument at least two. One effective way to overcome this difficulty is to define some symmetrized versions of the MZVs, which will be carried out in this chapter.

A multiple harmonic sum (MHS) is a finite partial sum of the infinite series in Eq. (1.8) defining the MZV ζ(s₁, …, s_d) for s₁, …, s_d ∈ ℕ (note that s₁ = 1 is allowed for partial sums). We have already encountered these numbers in previous chapters (see (3.18)). An alternating MHS (AMHS for abbreviation) is a finite partial sum of the infinite series defining a Euler sum, an alternating version of a MZV. All of these partial sums are rational numbers so it makes perfect sense to consider congruence properties modulo sufficiently large primes. This will be our first goal in this chapter. The second goal is to derive some identities using the binomial coefficients which will be useful to evaluate the MZVs and similar objects when the number of terms of the partial sums goes to infinity.

A general philosophy in studying congruences is to consider so-called van Hamme type congruences. One starts with an intriguing infinite series whose terms are given by rational numbers with some pattern and then, for suitable primes p, looks at the (p−1)-st partial sum modulo p, or even modulo higher powers of p which leads to super congruences.

We have seen from the preceding two chapters that the MHSs and MH⋆Ss not only have a lot of nice properties but also are closely intertwined with the MZVs and the MZ⋆Vs. In this chapter, we move on to study their q-analogs which again will have important applications in a number of results concerning the MZVs and the MZ⋆Vs.

The multiple zeta star values are a kind of variation of the MZVs. For any fixed positive integer d and s = (s₁, …, s_d) ∈ ℕ^d, s₁ ≥ 2, the multiple zeta star function is defined by

In this chapter we define some q-analogs of the multiple zeta functions and study their analytic properties. We will also consider some special values at non-positive integers. The theory of special values at positive integers will be developed in the next chapter. As before, throughout this chapter let q be a real number with 0 < q < 1.

There are quite a few different ways to define q-analogs of the MZVs and the MZ⋆Vs. We require that a good analog should not only deform into its classical counterpart when q → 1 but also satisfy the key relations that hold in the classical setting. Therefore, we seek for the q-analogs of the MZVs (q-MZVs for abbreviation) that obey some kind of double shuffle relations (DBSFs). On the other hand, for the q-analogs of the MZ⋆Vs (q-MZ⋆Vs for abbreviation), we choose those which enjoy interesting identities similar to the non-q version. In this chapter we present some of these and consider their most important properties.

In Chap. 2 we have studied mainly the analytic properties of the multiple polylog functions. In the following two chapters we turn to the algebraic side of the special values of these functions.

In this chapter, we concentrate on the special cases of CMZVs when the levels are two, four and six. The level two CMZVs are nothing but the (alternating) Euler sums. We start with a very useful online tool which can be used to check numerically if Euler sum identities are correct and also to detect ℚ-linear relations between them.

In this last chapter we shift our attention to one of the most active research areas in mathematical physics nowadays: the study of the Feynman integrals in the perturbative quantum field theory (QFT). For simplicity, we will consider only the four-dimensional case in Sections 15.1-15.4.

The following sections are included:

• Appendix A Key Concepts of Hopf Algebras
• Appendix B Some Useful Results from Lie Algebras
• Appendix C Basics of Hypergeometric Functions
• Appendix D Sample Computer Codes
• Appendix E Tables of Special Values
• Appendix F Answers to Some Exercises

The following sections are included:

• Bibliography
• List of Abbreviations
• List of Symbols
• Index