Multiple zeta values (多重ゼータ値)
Introduction to Algebra V (代数学概論V), 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] and [Z])
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.
References:
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] and [Z])
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], [Z])
- Multiple zeta values and their algebraic structure (harmonic & shuffle product) ([AK], [IKZ], [Z])
- Regularization of multiple zeta values ([IKZ])
- (Extended) Double shuffle relations ([IKZ])
- Families of linear relations among multiple zeta values ([B1], [IKZ], [Zh])
- 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]))
Materials
- Lecture notes: TBA
- Materials from the same course in 2020 can be found here. (The content will differ slightly)
Homework & Grading
The grading will be based on written homework assignments.
References:
- [AK] T. Arakawa, M. Kaneko, 多重ゼータ値入門 (Introduction to multiple zeta values (Japanese)). (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 and K. Ihara: Quasi-shuffle products revisited, J. Algebra 481 (2017), 293-326. (arXiv)
- [IKZ] K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math. 142 (2006), 307-338. (pdf)
- [W] M. Waldschmidt: Lectures on multiple zeta values. (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.
A good overview of research papers on multiple zeta values is the "Reference on multiple zeta values and Euler sum" by M. Hoffman.
Last update: 6th January 2025
\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}} }\]
\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}} }\]