发表机构
The Bishop’s School（主教学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出并构造了深度一和深度二下与经典欧拉和对应的有效欧拉和，并在附录中给出了深度二偶数权重的候选，证明了与金子-扎吉尔定义一致。
AI 中文摘要
欧拉和，也称为交错多重zeta值（MZVs），已被证明在数学和理论物理的许多研究领域中发挥重要作用。受金子（Kaneko）和扎吉尔（Zagier）关于多重zeta值的类似猜想的启发，作者提出了一个关于有限欧拉和空间与经典欧拉和空间模$\zeta(2)$-乘积的同构猜想。在本文中，我们明确构造了在深度一和深度二情况下，在该猜想下对应于经典欧拉和的那些有限元素（我们称之为有效欧拉和）。在附录中，我们通过扩展金子与扎吉尔关于二重zeta情形的启发式论证，提供了深度为二且权重为偶数时的另一组合理的有效欧拉和候选。基纳（Kina）最近独立地在多重zeta值设定中获得了一些相同的结果。特别地，利用他的一个结果，我们在附录中证明了金子与扎吉尔启发式定义的有效二重zeta值与我们的定义一致。
英文摘要
Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo $ζ(2)$-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.
Comments25 pages, with an appendix by R. Shi and J. Zhao
