Special values of the spectral zeta function for the noncommutative harmonic oscillator and modular forms

2009 Fiscal Year Final Research Report

• Project/Area Number: 20740021
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: University of the Ryukyus
• Principal Investigator: KIMOTO Kazufumi University of the Ryukyus, 理学部, 助教 (10372806)
• Project Period (FY): 2008 – 2009
• Keywords: 非可換調和振動子 / スペクトルゼータ関数 / 特殊値 / モジュラー形式

Research Abstract

We have studied the spectral zeta function associated to a differential operator called the noncommutative harmonic oscillator (NCHO in short). Each special value (i.e. the values at integral points) of the spectral zeta is expressed as a sum of a Riemann zeta value and certain terms (remainder terms) involving the structure parameter of NCHO, which is thought to reflect the noncommutativity of the NCHO. These remainder terms induce higher analogue of Apery-like numbers and some variants of multiple zeta values. We found a certain structure among the generating functions of these higher Apery-like numbers, as well as calculate explicitly the multiple zeta values associated to the remainder terms.

Research Products

• [Journal Article] A variation of multiple L-values arising from the spectral zeta function of the non-commutative harmonic oscillator. (2009)
  • Author(s): K. Kimoto, Y. Yamasaki
  • Journal Title: Proc. Amer. Math. Soc. 137 Pages: 2503-2515
  • Peer Reviewed
• [Presentation] 非可換調和振動子のスペクトルゼータ関数の特殊値について (2009)
  • Author(s): 木本一史
  • Organizer: 2009年度日本数学会秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Remarks] プレプリント
• [Remarks] K. Kimoto: Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator. arXiv:0901.0658
• [Remarks] K. Kimoto: Special value formula for the spectral zeta function of the non-commutative harmonic oscillator. arXiv:0903.5165
• [Remarks] 研究集会等での講演
• [Remarks] K. Kimoto: Arithmetic aspects of the non-commutative harmonic oscillator. K-Theory, Quadratic Forms and Number Theory Seminars, University College Dublin, 2008年10月
• [Remarks] K. Kimoto: Special values of the spectral zeta function of the non-commutative harmonic oscillator. Institute of Advanced Studies, University of Bologna, 2008年11月
• [Remarks] K. Kimoto: Arithmetics derived from the noncommutative harmonic oscillator. Casimir Force, Casimir Operators and the Riemann Hypothesis, 九大西新プラザ, 2009年11月
• [Remarks] K. Kimoto: Spectral anomalies in special values of the spectral zeta function associated to the non-commutative harmonic oscillator. Zetas and Limit Laws in Okinawa 2009, 沖縄コンベンションセンター, 2009年11月