Multiple zeta values

Content

Multiple zeta values are real numbers appearing in several areas of mathematics and theoretical physics. These numbers are natural generalizations of special values of the Riemann zeta function. They have been studied at least since Euler, who found many of their algebraic properties. Having been seemingly forgotten for more than 200 years, multiple zeta values were rediscovered by many mathematicians and theoretical physicists since the 1980s in several different contexts (modular forms, mixed Tate motives, quantum groups, moduli spaces of vector bundles, scattering amplitudes, resurgence theory, etc.). See the textbooks/lecture notes [AK], [BF], [W], [Zu] for more and better explanations on some of the topics we will (not) cover in this course.

Goals of the Course

​The goal of this course is to give an introduction to the theory of multiple zeta values, their variants and their connection to modular forms as given in [GKZ] and [B2]. For this, we will cover roughly the following topics

• The Riemann zeta-function and its special values ([AK], ​[BF], [W],[Zu])
• Multiple zeta values and their algebraic structure (harmonic & shuffle product) ([AK], [IKZ],[Zu])
• Regularization of multiple zeta values ([IKZ])
• (Extended) Double shuffle relations ([IKZ])
• Families of linear relations among multiple zeta values ([B1], [IKZ], [Zh],[Zu])
• Finite & Symmetric multiple zeta values
• Modular forms and their period polynomials ([GKZ], [Za])
• Connection of modular forms and multiple zeta values ([GKZ])
• Multiple Eisenstein series ([B2], [GKZ])
• q-analogues of multiple zeta values ([B2],[Zu])

Teaching Tips

The course is structured around exploring four interconnected "worlds" of mathematical objects: the real numbers (where classical MZVs live), the ring of "poor man's adeles" (for finite MZVs), holomorphic functions (modular forms and Eisenstein series), and q-series (q-analogues of MZVs). My goal is to illustrate the surprising and deep connections between these seemingly disparate areas. I emphasize not just the theoretical results but also the joy of discovery, encouraging students to use computational tools like PARI/GP to find patterns and relations for themselves, just as early mathematicians did. Ultimately, I want students to see how a single beautiful idea can weave together numerous branches of modern mathematics.

Course schedule

1. Course Overview & The Riemann zeta-function
2. Multiple zeta values
3. Finite & Symmetric multiple zeta values I
4. SFinite & Symmetric multiple zeta values II, Modular forms
5. Cusp forms & MZV, Quasimodular forms & sl2-algebras
6. Multiple Eisenstein series
7. q-analogues of MZV & Algebraic Setup I
8. Multiple Polylogarithm, Iterated integrals, Shuffle product
9. Stuffle product, Finite double shuffle relations, Quasi-shuffle products I
10. Quasi-shuffle products II
11. Linear maps induced from power series, Regularization, Extended double shuffle relations
12. Extended double shuffle relations II & Open Problems
13. Open Problems II & Finite/Symmetric MZV II
14. Finite/Symmetric MZV III
15. BTT-Philiosophy, Fourier expansion of multiple Eisenstein series.

Materials

• Lecture notes (Version 25, 28th August 2025)
• Homework: Homework 1, Homework 2, Homework 3
• Quasi-shuffle products playground (online tool)

Lecture videos (*previous version of this course)

• Introduction, Riemann zeta-function, and MZV I: Lecture1, Lecture2
• MZV II, Modular forms I, and q-analogues of MZV I: Lecture3, Lecture4
• Multiple polylogs, iterated integrals & finite double shuffle: Lecture5
• Quasi-shuffle algebras I: Lecture6
• Quasi-shuffle algebras II & Regularizations: Lecture7
• Extended double shuffle relations: Lecture8
• Seki-Yamamoto's connected sums & Ohno's relation: Lecture9
• Formal double zeta space I: Lecture10
• Formal double zeta space II: Lecture11
• Period polynomials I: Lecture12
• Multiple Eisenstein series: Lecture13
• Regularized multiple Eisenstein series: Lecture14
• double indexed q-analogues of MZV II: Lecture15

Evaluation methods

The grading will be based on written homework assignments.

Reference Book

• [AK] T. Arakawa, M. Kaneko, 多重ゼータ値入門 (Introduction to multiple zeta values) 🇯🇵 (pdf)
• [B1] H. Bachmann, Multiple zeta values and their relations (poster). (pdf)
• [B2] H. Bachmann, Multiple Eisenstein series and q-analogues of multiple zeta values. (pdf)
• [BF] J. Burgos, J. Frésan: Multiple zeta values: From numbers to motives. (pdf)
• [GKZ] H. Gangl, M.Kaneko, D. Zagier: Double zeta values and modular forms, in "Automorphic forms and zeta functions" World Sci. Publ., Hackensack, NJ (2006), 71-106. (pdf)
• [H] M. E. Hoffman: The algebra of multiple harmonic series, J. Algebra 194 (1997), 477-495. (pdf)
• [HI] M. E. Hoffman, K. Ihara: Quasi-shuffle products revisited, J. Algebra 481 (2017), 293-326. (arXiv)
• [IKZ] K. Ihara, M. Kaneko, D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math. 142 (2006), 307-338. (pdf)​
• [K] M. Kaneko: 有限多重ゼータ値（Finite multiple zeta values), RIMS Kôkyûroku Bessatsu B68, 175-190, (2017) 🇯🇵. (pdf)
• [M] H. Murahara: A study on relations among finite multiple zeta values, Doctoral thesis (2016). (pdf)
• [W] M. Waldschmidt: Lectures on multiple zeta values. (pdf)
• [XZ] C. Xu, J. Zhao: 多重 zeta 值及相关理论​ (Multiple Zeta Values and Related Theories) 🇨🇳​. (pdf)
• [Za] D. Zagier, Modular Forms of One Variable. (pdf)
• [Za2] D. Zagier, The 1-2-3 of Modular Forms. (pdf)
• [Zh] J. Zhao: Multiple zeta functions, multiple polylogarithms and their special values, New Jersey: World Scientific, (2016)​.
• [Zu] W. Zudilin: Multiple zeta values, tasting notes (2025). (pdf)

M. Hoffman's "Reference on multiple zeta values and Euler sum" provides a good overview of research papers on multiple zeta values.

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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