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

我们研究了与称为非对易谐振子(简称NCHO)的微分算子有关的谱Zeta函数。谱Zeta的每个特殊值(即积分点处的值)被表示为Riemann Zeta值与涉及NCHO的结构参数的某些项(余项)之和，这被认为反映了NCHO的非对易性。这些余数项导致了Apery类数和多个Zeta值的一些变体的更高的相似性。我们发现了这些高阶Apery数的母函数之间的某种结构，并显式地计算了与余项相关的多个Zeta值。

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.