Multiple zeta values (多重ゼータ値)
Introduction to Algebra V (代数学概論V) / Algebra I (代数学I), Nagoya University, Spring 2025
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.
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
Materials
Homework & Grading
The grading will be based on written homework assignments. Please be sure to be part of the TACT group for this course.
宿題の提出は日本語でも構いません。その場合は、読みやすく書いてください。
Place & Time: Every week, Thursday 3rd period (13:00 - 14:30) in Room 509 (Math. Building)
Course schedule (Tentative):
The following gives a tentative overview of the topics we will cover each week.
Week 01 (04/07-04/13): Course Overview & The Riemann zeta-function
Week 02 (04/14-04/20): Multiple zeta values
Week 03 (04/21-04/27): Finite & Symmetric multiple zeta values I
Week 04 (04/28-05/04): Finite & Symmetric multiple zeta values II, Modular forms
Week 05 (05/05-05/11): Cusp forms & MZV, Quasimodular forms & sl2-algebras
Week 06 (05/12-05/18): Multiple Eisenstein series
Week 07 (05/19-05/25): TBA
Week 08 (05/26-06/01): TBA
Week 09 (06/02-06/08): No Lecture (名大祭)
Week 10 (06/09-06/15): TBA
Week 11 (06/16-06/22): TBA
Week 12 (06/23-06/29): No Lecture (Conference)
Week 13 (06/30-07/06): TBA
Week 14 (07/07-07/13): TBA
Week 15 (07/14-07/20): TBA
Week 16 (07/21-07/27): TBA
Week 17 (07/28 - 08/03): TBA
References:
M. Hoffman's "Reference on multiple zeta values and Euler sum" provides a good overview of research papers on multiple zeta values.
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.
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])
Materials
- Lecture notes (Version 15, 9th May 2025)
- Course information
- Homework: Homework 1
- Materials from a similar course in 2020 can be found here. (The content will differ)
Homework & Grading
The grading will be based on written homework assignments. Please be sure to be part of the TACT group for this course.
宿題の提出は日本語でも構いません。その場合は、読みやすく書いてください。
Place & Time: Every week, Thursday 3rd period (13:00 - 14:30) in Room 509 (Math. Building)
Course schedule (Tentative):
The following gives a tentative overview of the topics we will cover each week.
Week 01 (04/07-04/13): Course Overview & The Riemann zeta-function
Week 02 (04/14-04/20): Multiple zeta values
Week 03 (04/21-04/27): Finite & Symmetric multiple zeta values I
Week 04 (04/28-05/04): Finite & Symmetric multiple zeta values II, Modular forms
Week 05 (05/05-05/11): Cusp forms & MZV, Quasimodular forms & sl2-algebras
Week 06 (05/12-05/18): Multiple Eisenstein series
Week 07 (05/19-05/25): TBA
Week 08 (05/26-06/01): TBA
Week 09 (06/02-06/08): No Lecture (名大祭)
Week 10 (06/09-06/15): TBA
Week 11 (06/16-06/22): TBA
Week 12 (06/23-06/29): No Lecture (Conference)
Week 13 (06/30-07/06): TBA
Week 14 (07/07-07/13): TBA
Week 15 (07/14-07/20): TBA
Week 16 (07/21-07/27): TBA
Week 17 (07/28 - 08/03): TBA
References:
- [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)
- [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.
Last update: 9th May 2025
\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \cdots m_r^{k_r}} }\]
\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \cdots m_r^{k_r}} }\]