Depth-graded motivic multiple zeta values

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: ℌ0 → R from a graded rational vector space ℌ0 generated by the “admissible words” of the noncommutative polynomial algebra Q〈x,y〉. Now ℌ0 admits two (commutative) products making ζ a homomorphism-the shuffle product and the “harmonic” product. The latter makes ℌ0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.

The multiple zeta values (MZVs) have been studied extensively in recent years. Currently there exist a few different types of $q$-analogs of the MZVs ($q$-MZVs) defined and studied by mathematicians and physicists. In this paper, we give a uniform treatment of these $q$-MZVs by considering their double shuffle relations (DBSFs) and duality relations. The main idea is a modification and generalization of the one used by Castillo Medina et al. who have considered the DBSFs of a special type of $q$-MZVs. We generalize their method to a few other types of $q$-MZVs including the one defined by the author in 2003. With different approach, Takeyama has already studied this type by "regularization" and observed that there exist $\mathbb Q$-linear relations which are not consequences of the DBSFs. He also discovered a new family of relations which we call the duality relations in this paper. This deficiency of DBSFs occurs among other types, too, so we generalize the duality relations to all of these values and find that there are still some missing relations. This leads to the most general type of $q$-MZVs together with a new kind of relations called $\bf P$-$\bf Q$ relations which are used to lower the deficiencies further. As an application, we will confirm a conjecture of Okounkov on the dimensions of certain $q$-MZV spaces, either theoretically or numerically, for the weight up to 12. Some relevant numerical data are provided at the end.

We study in some detail the very important class of periods called multiple zeta values (MZV). These are periods of mixed Tate motives, which we discussed in Sect. 6.4. Multiple zeta values are in fact periods of unramified mixed Tate motives, a full subcategory of all mixed Tate motives.

A generalized regularization theorem and Kawashima's relation for multiple zeta values

Scattering amplitudes which describe the interaction of physical states play an important role in determining physical observables. In string theory the physical states are given by vibrations of open and closed strings and their interactions are described (at the leading order in perturbation theory) by a world-sheet given by the topology of a disk or sphere, respectively. Formally, for scattering of N strings this leads to N-3-dimensional iterated real integrals along the compactified real axis or N-3-dimensional complex sphere integrals, respectively. As a consequence the physical observables are described by periods on M_{0,N} - the moduli space of Riemann spheres of N ordered marked points. The mathematical structure of these string amplitudes share many recent advances in arithmetic algebraic geometry and number theory like multiple zeta values, single-valued multiple zeta values, Drinfeld, Deligne associators, Hopf algebra and Lie algebra structures related to Grothendiecks Galois theory. We review these results, with emphasis on a beautiful link between generalized hypergeometric functions describing the real iterated integrals on M_{0,N}(R) and the decomposition of motivic multiple zeta values. Furthermore, a relation expressing complex integrals on M_{0,N}(C) as single-valued projection of iterated real integrals on M_{0,N}(R) is exhibited.

Multiple series connected to Hoffman's conjecture on multiple zeta values

The Ohno relation is one of the most celebrated results in the theory of multiple zeta values, which are given in the form of iterated integrals from 0 to 1. In a previous paper, the authors generalized the Ohno relation to regularized multiple zeta values, which are given in the form of non-admissible iterated integrals from 0 to 1. Meanwhile, Takeyama proved an analogue of the Ohno relation for refined symmetric multiple zeta values, which are iterated integrals from 0 to 0. In this paper, we generalize Takeyama’s result to regularized refined symmetric multiple zeta values, which are non-admissible iterated integrals from 0 to 0.

On three general forms of multiple zeta(-star) values

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings {1, 3, . . . , 1, 3} and {2, . . . , 2} is a rational multiple of a power of π². Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring A₁ vanish. In this paper, we partially lift Saito-Wakabayashi’s theorem from A₁ to A₂. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in A₂ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.

Relations of multiple zeta values and their algebraic expression

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras introduced by the second author. If the number of variables involved is “not large” in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over ${\mathbb{F}}_q$. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of linear relations between Thakur’s multiple zeta values.

We show that the single trace heterotic N-point tree-level gauge amplitude ANHET can be obtained from the corresponding type I amplitude ANI by the single-valued (sv) projection: ANHET=sv(ANI). This projection maps multiple zeta values to single-valued multiple zeta values. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Similar relations between open and closed string amplitudes or amplitudes of different string vacua can be established. As a consequence the α′-expansion of a closed string amplitude is dictated by that of the corresponding open string amplitude. The combination of single-valued projections, Kawai–Lewellen–Tye relations and Mellin correspondence reveal a unity of all tree-level open and closed superstring amplitudes together with the maximally supersymmetric Yang–Mills and supergravity theories.

Non-vanishing of multiple zeta values for higher genus curves over finite fields

Multiple zeta values are the numbers defined by the convergent seriesζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), wheres1,s2,…,skare positive integers withs1>1. Fork≤n, letE(2n,k)be the sum of all multiple zeta values with even arguments whose weight is2nand whose depth isk. The well-known resultE(2n,2)=3ζ(2n)/4was extended toE(2n,3)andE(2n,4)by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbersE(2n,k)and then gave a direct formula forE(2n,k)for arbitraryk≤n. In this paper we apply a technique introduced by Granville to present an algorithm to calculateE(2n,k)and prove that the direct formula can also be deduced from Eisenstein's double product.

In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℂ</mml:mi> <mml:mi>r</mml:mi> </mml:msup></mml:math> with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the shuffle product. We give a detailed analysis about the depth structure of continuous multiple zeta values. There are also sum formulas for continuous multiple zeta values. Lastly we calculate some special continuous multiple zeta values in terms of special values of multiple polylogarithms.