零和理论是组合数论中热门的研究领域,应用非常广泛, 相关的结果被应用到许多其他数学领域，这包括代数数论、离散几何、图论及Ramsey理论等。零和理论的主题是研究零和序列，也就是在加法交换群G中，元素之和为零元的序列。那么对于各种具有不同性质的零和序列的研究是零和理论中的一个重要课题，而计算对应产生的不变量成为零和理论研究中的重要工作。本项目主要研究有关cross number的几个不变量;在群G上，一个序列S的cross number 定义为S中所有元素的阶的倒数之和。本项目从零和理论中几个核心的常数与零和基本等式出发，综合使用来自代数、图论、数论、组合数学以及概率的工具和方法对不变量t(G)和\eta (G) 进行研究。希望通过本项目的探索，取得一些较好的研究结果，进而推进和丰富零和理论的研究。

Zero-sum theory is a modern field of Combinatorial Number Theory and its content is vast. The concerning results are also applied widely in many different scientific subjects including Algebraic Number Theory, Discrete Geometry, Graph Theory, and Ramsey Theory, etc. The main theme of zero-sum theory is to study the zero-sum sequence which is a sequence consisting of elements in an additive finite abelian group G with the sum zero. The various properties of zero-sum sequences are very interesting and became the important subject of zero-sum theory，and to calculate the value of the relevant invariants is an important job. In a short word, this project is to study some invariants concerning the cross number of G. The cross number of a sequence S over a group G is defined as the sum of all the inverse of the order of its elements. Based on the kernel constants and fundamental zero-sum equalities, this project is to study t(G) and \eta (G) . We hope to get some good results, and improve the research of zero-sum theory.