本项目要研究自然数表为三角形数线性组合的方法数公式，揭示其与自然数表为平方数线性组合方法数之间的联系，解决申请人提出的一些猜想。此外，研究Ramanujan提出的一些关于eta函数乘积和Dirichlet级数Euler乘积的一些猜想，研究与eta函数乘积或eta函数商有关的可乘函数性质，揭示eta函数乘积与Jacobsthal和之间的联系，努力解决申请人的一些相关猜想。

The purpose of the project is to study the number of representations of a given positive integer as a linear combination of several triangular numbers, reveal the connections between the number of representations of n as a linear combination of several triangular numbers and the number of representations of m as a linear combination of several squares， and prove some conjectures of the applicant. In addition, we try to confirm Ramanujan’s conjectures on products of eta functions and Euler products for certain Dirichlet series, investigate the properties of multiplicative functions related to eta products or eta quotients, reveal the connections with Jacobsthal sums and try to solve the applicant’s related conjectures.