**Multiple zeta values and modular forms**

Topics in Mathematical Science III (Nagoya University, Spring 2020)

**(This page will be continuously updated before and during the course. Please check for updates regularly.)**

**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 and to explain their connection to modular forms as given in [GKZ]. For this, we will cover the following topics (tentative)

- 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])
- Modular forms and their period polynomials ([GKZ], [Za])
- Connection of modular forms and multiple zeta values ([GKZ])
- Multiple Eisenstein series ([B2], [GKZ])

**Materials**

- Course Information (coming soon)
- Homework (coming soon)
- Lecture notes (coming soon)

**Grading**

The grading will be based on written homework assignments.

**Lecture schedule:**

The lectures will take place

**Friday**at 8:45 -10:15. (Room is not decided yet)

First lecture: 17th April 2020 (This might change due to the Corona situation)

**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) - [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.

Last update: 15th March

\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}} }\]

\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}} }\]