拉马努金西塔函数及其应用

As an important class of arithmetic functions, theta functions, which were studied systematically by Jacobi first and then investigated extensively by Ramanujan, have become a hot research subject between combinatorics and number theory. Theta function identities occur most commonly in the theory of theta functions, and play a key role in many mathematical areas, such as representations by quadratic forms and the theory of integer partitions..We will mainly investigate the theta function identities using the theory of q-series and their applications to combinatorics and number theory. The research focuses on the following topics:.(1)representation of integers by sums of square numbers and by sums of triangular numbers, mainly including the conjecture of Zhihong Sun, the q-series proof of Liouville's assertation, and representation of integers by variable with congruence conditions;.(2)the arithmetic properties of partition functions, especially the arithmetic properties of those related to mock theta functions and the refinement of Ramanujan's classical congruence that p(7n+5) is divisible by 7;.(3)studying the Lambert series expansions for the multiplications of Borweins' theta functions.

西塔函数是一类重要的算术函数，由雅可比最先系统研究，后经拉马努金更加全面探讨，现已成为组合数学与数论交叉研究的一个热门课题。西塔函数恒等式作为西塔函数理论的重要组成部分，在许多数学领域中都发挥着重要作用，如在数的二次型表示和整数分拆理论方面。.本项目主要利用q-级数理论方法来研究西塔函数恒等式及其在组合数学与数论中的应用，拟从三个方面来展开研究：.(1)数的平方和表示与数的三角数和表示，主要包括孙智宏的猜想、刘维尔结论的q-级数证明、变量附有同余条件的数的表示；.(2)分拆函数的算术性质，重点研究涉及mock西塔函数的相关函数的算术性质和寻找拉马努金经典同余式p(7n+5)被7整除的细分；.(3)研究Borweins西塔函数乘积的Lambert级数展开式。

拉马努金西塔函数在q-级数、分拆理论以及数的二次型表示等方面发挥着重要作用。本项目主要的研究内容有：.（1）解决了孙智宏教授的若干关于k个三角数和表示成k个平方数和的线性组合的猜想，其中k=3，4；.（2）研究两个与mock西塔函数相关的分拆函数的算术性质，给出一个约束相邻部分差的带杠分拆函数的模5与模27的无穷同余子列，研究统计量lrank的相关生成函数及同余性质以及k-colored分拆的Andrews-Beck型同余式；.（3）研究最大部分与最小部分差有界的几类分拆的生成函数，给出Andrews等人定义的2-measure生成函数的细分以及Campbell定义的几乎自共轭分拆函数的生成函数；.（4）研究若干特殊无穷乘积的子列的生成函数。.在项目运行期间，项目组成员总共发表学术论文15 篇，其中12篇已正式发表，3篇已在线发表。