多重zeta值及相关理论  

Multiple Zeta Values and Related Theories

作　　者：徐策 赵健强 XU Ce;ZHAO Jianqiang(School of Mathematics and Statistics,Anhui Normal University,Wuhu,Anhui,241002,P.R.China;Department of Mathematics,The Bishop's School,La Jolla,CA 92037,USA)

机构地区：[1]安徽师范大学数学与统计学院,安徽芜湖241002 [2]毕夏普学校数学系,美国拉荷亚,CA92037

出　　处：《数学进展》2025年第2期315-342,共28页Advances in Mathematics(China)

基　　金：国家自然科学基金(No.12101008)。

摘　　要：近年来,多重zeta值及相关理论已被广泛地应用于量子物理学、纽结理论、数论、非交换几何等学科中,对多重zeta值相关问题的研究已成为数学和物理学中非常活跃的一个领域.至今,该领域已取得了许多有趣且深刻的结果,但仍然有许多公开的问题和猜想有待解决.本文我们将系统地阐述多重zeta值及相关变形的算术、代数、组合及几何性质,特别是对多重zeta值及其奇偶变形(如:多重t-zeta值,多重T-zeta值,多重S-zeta值,多重M-zeta值及其交错形式等)、q-模拟多重zeta值、motive理论和罗巴代数理论等相关理论的关联进行综述.In recent years,the multiple zeta values and related variants have been widely used in quantum physics,knot theory,number theory,non-commutative geometry,etc.Consequently,the research on the relevant problems has become a very active field in mathematics and physics.Many interesting and profound results have been achieved in this field so far,however,there are still many open problems and conjectures to be solved.This survey will systematically introduce the arithmetic,algebraic,combinatoric and geometric properties of these fascinating objects.In particular,we will expound on the connections between the multiple zeta values and some of their odd and even variants(such as multiple t-values,multiple T-values,multiple S-values,multiple M-values and their alternating versions) and their q-analogs and explain how the theory of motives and Rota-Baxter algebras are applied in this framework.

关 键 词：多重zeta值 奇偶变形 对偶关系 双洗牌关系 q-模拟多重zeta值 motive理论 罗巴代数 

分 类 号：O156[理学—数学]