Multiple Zeta Values

• Multiple zeta values are nested series that generalize the Riemann zeta function, serving as fundamental objects with rich arithmetic and combinatorial properties.
• They exhibit intricate algebraic structures through double shuffle relations and Lyndon word combinatorics, linking them to modular forms and period integrals.
• Their study advances motivic Galois theory and quantum field theory applications while presenting open challenges in convergence and explicit identity formulations.

Multiple zeta values (MZVs) are special values of nested series generalizing the Riemann zeta function to several complex variables. They play a fundamental role at the interface of number theory, algebraic geometry, and mathematical physics, revealing deep structures connected with period integrals, motivic Galois groups, and quantum field theory. The paper of their algebraic, combinatorial, and arithmetic properties has led to a rich interconnection with multiple polylogarithms, modular forms, and the theory of motives.

1. Definitions and Core Properties

A (classical) multiple zeta value is defined for integers $k_1, \ldots, k_d$ with $k_1 \geq 2$, as

$$\zeta(k_1, \ldots, k_d) = \sum_{n_1 > n_2 > \cdots > n_d \geq 1} \frac{1}{n_1^{k_1} n_2^{k_2} \cdots n_d^{k_d}}$$

where $d$ is the depth and $k_1 + \dots + k_d$ the weight.

These values generalize the Riemann zeta values $\zeta(k)$ ($d=1$). In the case of $k=2$, Euler showed $\zeta(2) = \pi^2 / 6$, and for even $k$, established the formula

$\zeta(2n) = \frac{|B_{2n}|}{2(2n)!}(2\pi)^{2n}$

with $B_{2n}$ the Bernoulli numbers, reflecting a link between special values and periods of algebraic varieties (Dupont, 2021).

MZVs at odd arguments and with multiple indices are more mysterious; many conjectures concern their algebraic and transcendental nature. Notably, the series for MZVs often converge slowly, making evaluation and relation discovery challenging (Akhilesh, 2019).

2. Algebraic Structures and Double Shuffle Relations

MZVs naturally inhabit an algebra over $\mathbb{Q}$, with the so-called double shuffle structure arising from two types of product expansions:

• The stuffle product results from multiplying the underlying series representations (quasi-shuffle): it expresses $\zeta(m)\zeta(n)$ as a sum over MZVs of depth two or less.
• The shuffle product arises from interpreting MZVs as iterated integrals: it encodes algebraic relations among their expressions via integral shuffling.

These double shuffle relations generate a complex family of algebraic dependencies among MZVs; understanding which are fundamental is a central topic (1207.1735). Lyndon word combinatorics play a key role: the algebra is conjecturally generated by MZVs whose index sequences are Lyndon words in {2,3} (the “2–3 conjecture”) (1207.1735).

Zagier's conjecture posits that the dimension $d_n$ of the weight-$n$ space of $\mathbb{Q}$-linear MZVs satisfies the recurrence $d_n = d_{n-2} + d_{n-3}$ with $d_2 = d_3 = 1$ (Dupont, 2021, 1207.1735).

3. Motivic and Galois-Theoretic Frameworks

Recent decades have seen the emergence of a motivic and Galois-theoretic lens. MZVs are interpreted as periods of mixed Tate motives over $\mathbb{Z}$ (Dupont, 2021).

• Motivic MZVs $\zeta^{\mathrm{mot}}(k_1,\dots,k_d)$ are formal objects in a category equipped with a pro-unipotent Galois group acting on them; their complex-valued periods recover the classical MZVs.
• A period conjecture states that all $\mathbb{Q}$-linear relations among classical MZVs come from motivic relations—if true, this tightly constrains possible identities.
• The algebra generated by motivic MZVs is conjecturally isomorphic to a free polynomial algebra in specific generators, mirroring the dimension recurrence above and enabling a Galois–theoretic approach to MZV structure (Dupont, 2021, Charlton et al., 2022).

4. Combinatorics, Explicit Formulas, and Canonical Decompositions

Efforts to express MZVs in terms of rapidly converging series, such as multiple Apéry-like sums (generalizing Apéry's sequences for $\zeta(2)$ and $\zeta(3)$), have intensified (Akhilesh, 2019). Every MZV can be written as an explicit $\mathbb{Z}$-linear combination of Apéry-like sums; the paper (Akhilesh, 2019) describes a canonical map $d$ associating self-dual classes of admissible index compositions to such combinations. This $d$ map is characterized by strict normalization properties and a recursion reflecting binary word decompositions of indices.

The combinatorics of the underlying compositions, together with a modified stuffle product (where the first components are added and the rest “stuffled”), reveal intricate relations among MZVs and Apéry-like sums, including a direct proof of Zagier's formula for the family $\zeta(2,\ldots,2,3,2,\ldots,2)$. The convergence of Apéry-like sums is typically much faster than for MZVs, making them especially well-suited for computations (Akhilesh, 2019).

5. Filtration, Period Polynomials, and Lie Algebraic Aspects

Three filtration concepts are central to the structure of MZVs (Charlton et al., 2022):

• Depth filtration: Graded by the length of the index tuples (the number of nested summations).
• Coradical filtration: Arises from the coalgebra structure on motivic MZVs, isolating indecomposable elements.
• Block filtration: A combinatorial grading by the number of “blocks” in the iterated integral word. This filtration matches the coradical filtration, resulting in a well-behaved associated graded Lie algebra.

Quadratic relations observed in the depth-graded algebra originate from the period polynomials of cusp forms for $SL_2(\mathbb{Z})$; however, in the block filtration, these “defects” disappear, making the algebra simpler and more amenable to Lie theoretic and motivic analysis. The kernel of the Ihara bracket acting on the block-graded Lie algebra aligns exactly with the space of these period polynomials (Charlton et al., 2022).

Evaluations such as

$$\zeta^{l}(\{2\}^{a}, 4, \{2\}^{b}) = (-1)^{a+b} \Big[ -4 \zeta^{l}(2a+2,2b+2) + \cdots \Big] \pmod{\text{products}}$$

demonstrate explicit relationships between single and multiple zeta values and their motivic and block-graded projections (Charlton et al., 2022).

6. Applications in Modular Forms, Function Field Arithmetic, and Physics

MZVs have far-reaching implications:

• Modular Forms and Period Polynomials: The interplay between MZVs and modular form period polynomials provides an explanation for the structure of quadratic relations in depth-graded MZVs (Charlton et al., 2022).
• Function Field Analogues: Over positive characteristic, multiple zeta values attached to function fields depend on the constant field. Recent advances using t-motives and Papanikolas's theorem show that MZVs from different constant fields are algebraically independent; no algebraic relations mix MZVs from different constants (Matsuzuki, 29 Jun 2024).
• Quantum Field Theory: Feynman integral computations in certain $2$-dimensional theories yield periods that are MZVs, with the motivic structure offering a “cosmic Galois group” action on amplitudes (Dupont, 2021).

7. Further Directions and Open Problems

While much is now understood about producing and relating MZVs via combinatorial, motivic, and period-theoretic machinery, some aspects remain mysterious:

• The structure of relations among Apéry-like sums is only partially understood, and a full description—possibly relating to motivic depth filtration—is open (Akhilesh, 2019).
• The conjecture that all MZVs with indices in $\{2,3\}$ can be written as a $\mathbb{Q}$-linear combination of a comparatively small number of Apéry-like sums with entries in $\{1,2,3\}$ and bounded odd part is supported but not proven (Akhilesh, 2019).
• The period conjecture, predicting that all $\mathbb{Q}$-linear relations among MZVs arise motivically, remains open.

In conclusion, the theory of multiple zeta values now stands at a confluence of explicit computations, deep motivic ideas, and broad applications—both theoretical and computational. Their paper continues to drive advancements in transcendence theory, the understanding of period Galois symmetries, and connections to arithmetic geometry and mathematical physics.