**Multiple zeta values and modular forms**

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

**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] 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])
- 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 & Exercises: Multiple zeta values & modular forms (ver. 5.4)
- Lecture videos Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11, Lecture 12, Lecture 13, Lecture 14, Lecture 15
- Zoom meeting notes: Meeting 2 (05/01), Meeting 3 (05/08), Meeting 4 (05/15), Meeting 5 (05/22), Meeting 6 (05/29), Meeting 7 (06/05), Meeting 8 (06/12), Meeting 9 (06/19), Meeting 10 (06/26), Meeting 11 (07/03), Meeting 12 (07/10), Meeting 13 (07/17), Meeting 14 (07/24), Meeting 15 (07/31), Meeting 16 (08/07)
- Quasi-shuffle products playground (online tool)
- Multiple Eisenstein series and their Fourier coefficients (recent talk slides related to this lecture)

**Lecture schedule**

**:**

Each week we will have a Zoom meeting at

**Friday 16:00-17:00**(Tokyo time) to discuss the current materials & exercises. The meetings ID and password are available in the NUCT course page. If you are not in the NUCT course and you want to join these meetings please write me an email.

The following gives a tentative overview of the topics we will cover each week.

Week 01 (04/20-04/26): Introduction, Riemann zeta-function, and MZV I (Subsection 1.1 & 1.2, Videos: Lecture 1, Lecture 2)

Week 02 (04/27-05/10): MZV II, Modular forms I, and q-analogues of MZV I (Subsection 1.3 & 1.4, Videos: Lecture 3, Lecture 4)

Golden week

Week 03 (05/11-05/17): Multiple polylogs, iterated integrals & finite double shuffle (Subsections 2.1+2.2, Video: Lecture 5)

Week 04 (05/18-05/24): Quasi-shuffle algebras I (Subsection 2.3, Video: Lecture 6)

Week 05 (05/25-05/31): Quasi-shuffle algebras II & Regularizations (Subsections 2.3 & 2.4, Video: Lecture 7)

Week 06 (06/01-06/07): Extended double shuffle relations (Subsections 2.4 & 3.1, Video: Lecture 8)

Week 07 (06/08-06/14): Seki-Yamamoto's connected sums & Ohno's relation (Subsections 3.2 & 3.3, Video: Lecture 9)

Week 08 (06/15-06/21): Formal double zeta space I (Subsection 4.1, Video: Lecture 10)

Week 09 (06/22-06/28): Formal double zeta space II (Subsection 4.1 & 4.2, Video: Lecture 11)

Week 10 (06/29-07/05): Period polynomials I (Subsection 4.3, Video: Lecture 12)

Week 11 (07/06-07/12): Period polynomials II (Subsection 4.3, Also see Meeting 12 (07/10))

Week 12 (07/13-07/19): Multiple Eisenstein series (Subsection 5.1, Video: Lecture 13)

Week 13 (07/20-07/26): Regularized multiple Eisenstein series (Subsection 5.2, Video: Lecture 14)

Week 14 (07/27-08/02): double indexed q-analogues of MZV I (Subsection 5.3, Also see Meeting 15 (07/31))

Week 15 (08/03-08/09): double indexed q-analogues of MZV II (Subsection 5.3, Video: Lecture 15)

**Homework & Grading**

**If you want to get credits for this course I expect you to join the Zoom lectures each Friday.**- The grading will be based on written homework assignments. During the course, we will provide Exercises (at the end of the lecture notes), which you need to hand in until the
**middle of****August**2020. - It is not decided yet how many Exercises there will be, but the idea is to give roughly 1-2 Exercises each week, which can be found at the end of the lecture notes or above.
- Please feel free to hand in Exercises separately before the deadline to get feedback as early as possible. (i.e. after finishing one Exercise please feel free to send it directly instead of sending the solutions of all exercises at once on August 9th)
- Please hand in your homework by email (henrik.bachmann (at) math.nagoya-u.ac.jp) as pdf files.

**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: 27th May 2021

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

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