q-analogues and finite multiple zeta values
​Topics in Mathematical Science III (Nagoya University, Spring 2022)

  • This lecture will be face-to-face.
  • For the homework submission we will use NUCT. Please make sure to be in this NUCT group. Also students who just want to join this lecture as a listener are welcome to join the NUCT group. 

Content (tentative)
The goal of this lecture is to explain the results obtained in the work [BTT]. There the authors give a new approach of connecting q-analogues of multiple zeta values and finite multiple zeta values. In this course, we will give an introduction to multiple zeta values, their q-analogues and their finite analogues. In particular, we will study their algebraic setup using quasi-shuffle algebras. 
  • Introduction to multiple zeta values ([AK], [B], [BF], [W], [Zh])
  • q-analogues of multiple zeta values ([B], [Br] , [T])
  • finite multiple zeta values ([K], [M], [Zh])
  • regularized and symmetric multiple zeta values
  • The Kaneko-Zagier conjecture
  • quasi-shuffle products & algebraic structure of q-analogues and finite MZV ([HI], [T])
  • The "BTT-Philosophy": Connection of q-analogues and finite MZV and Q-MZV ([BTT], [TT])
    ​
Materials​ Lecture schedule: 
The current plan is that the lectures will be face-to-face each Friday in Room 多453  (Math building) in the first period (8:45 - 10:15).  
First lecture: 15th April 2022,  Room 多453

Homework & Grading
  • If you want to get credits for this course I expect you to attend the lectures.  
  • The grading will be based on written homework assignments.

​Course schedule (Tentative): 
The following gives a tentative overview of the topics we will cover each week. 
Week 01 (04/11-04/17): Overview & Introduction to multiple zeta values
Week 02 (04/18-04/24):  Finite and symmetric multiple zeta values I 
Week 03 (04/25-05/01):  Golden week starts. No lecture on 29th April (Shōwa Day​)
Week 3.5 (05/02-05/08): Finite and symmetric multiple zeta values II 
Week 04 (05/09-05/15):  Kaneko-Zagier conjecture and the BTT-philosophy I
Week 05 (05/16-05/22):  No lecture (Conference: 多重ゼータ値の諸相)
Week 06 (05/23-05/29):  BTT-philosophy II & Quasi-shuffle products
Week 07 (05/30-06/05):  Quasi-shuffle products II & Multiple Polylogarithms 
Week 08 (06/06-06/12):  Regularization & Double shuffle relations for q-analogues I
Week 09 (06/13-06/19): Double shuffle relations for q-analogues II
Week 10 (06/20-06/26): - cancelled -
Week 11 (06/27-07/03): Double shuffle relations for q-analogues III & FMZV III  
Week 12 (07/04-07/10): Symmetric MZV
Week 13 (07/11-07/17): BTT-philosophy III
Week 14 (07/18-07/24): BTT-philosophy IIII & Q-MZV
Week 15 (07/25 - 07/31): - No lecture -

References
A more detailed list of references will be part of the lecture notes. 
  • [AK] T. Arakawa, M. Kaneko, 多重ゼータ値入門 (Introduction to multiple zeta values (Japanese)). (pdf)
  • [B] H. Bachmann: Multiple zeta values & modular forms (ver. 5.4), lecture notes (2021). 
  • [Br] B. Brindle: A unified approach to qMZVs, preprint (2021). (arxiv) 
  • [BF] J. Burgos, J. Frésan: Multiple zeta values: From numbers to motives, Clay Mathematics Proceedings. (pdf)
  • [BTT] H. Bachmann, Y. Takeyama, K. Tasaka: Cyclotomic analogues of finite multiple zeta values, Compositio Math. 154 (12), 2701-2721. (arxiv)
  • [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)
  • [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)
  • [T] Y. Takeyama: Derivations on the algebra of multiple harmonic q-series and their applications, The Ramanujan Journal 52,  41-65. (2020). (arxiv)
  • [TT] Y. Takeyama, K. Tasaka: Supercongruences of multiple harmonic q-sums and generalized finite/symmetric multiple zeta values, preprint (2020). (arxiv)
  • [W] M. Waldschmidt: Lectures on multiple zeta values. (pdf)
  • ​[Zh] J. Zhao: Multiple zeta functions, multiple polylogarithms and their special values, New Jersey: World Scientific, 2016​.
Last update: 21th July 2022
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\[ \color{gray}{ \zeta(k_1,\dots,k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}} }\] ​