类Dedekind和的相关问题研究

The studies on Dedekind sums occupy a pivotal position in analytic number theory, modular forms, algebraic number theory, combinatorial geometry, topology and algorithmic complexity. And many famous number theoretic problems such as transformation law for Dedekind eta function, class number formula, lattice point problems, group action on manifolds and pseudo-random number generators, are closely related to them. The main aim of the project is to study the upper bounds, high power mean and weighted mean of Hardy sums and Cochrane sums, which are quite analogous to Dedekind sums. To be more specific, by choosing some important exponential sums after deep some considerations, the weighted sums involving Hardy sums and Kloosterman sums with Dirichlet character or the two-term exponential sums with Dirichlet character are to be studied. The incomplete Cochrane sum is to be generalized to the high dimension case and the individual upper bound is to be obtained. And the high dimensional incomplete Cochrane sum weighted by the error term occurred in the high dimensional D.H. Lehmer problem is also to be studied. What’s more, the nontrivial conditions when two Hardy sums (or Cochrane sums) are equal are to be explored. Many techniques in number theory are to be applied, through which the distributive properties of these sums are revealed, and the theory involving them is complemented and enriched.

关于Dedekind和相关问题的研究在解析数论、模形式、代数数论、组合几何、拓扑学以及算术复杂性研究中占有举足轻重的地位，并和Dedekind η-函数的转化公式、类数、格点问题、群在流形上的作用以及随机数生成器等问题密切相关。本项目主要针对一些类Dedekind和，如Hardy和、Cochrane和的上界估计、高次均值、加权均值等进行研究。具体来说，拟通过巧妙选取一些重要指数和，如带特征Kloosterman和、带特征二项指数和等，给出Hardy和加权均值的一系列确切计算公式；拟研究不完整Cochrane和的高维推广，并给出其单个上界估计；拟获取高维不完整Cochrane和与高维D.H. Lehmer问题中误差项加权均值的渐近公式；拟建立两个Hardy和，或者两个Cochrane和相等的非平凡条件等。本项目将结合转化、选参、加权等技巧，力争揭示这些和式的均值分布规律，补充和发展相关理论。

关于Dedekind和相关问题的研究在解析数论、模形式、代数数论、组合几何、拓扑学以及算术复杂性研究中占有举足轻重的地位，并和Dedekind η-函数的转化公式、类数、格点问题、群在流形上的作用以及随机数生成器等问题密切相关。本项目主要针对一些类Dedekind和，如Hardy和、Poly-Dedekind和、Poly-Dedekind type DC和的混合均值、互反公式等进行了研究。具体来说，利用Gauss和的性质及Dirichlet L-函数的均值定理得到了Hardy和S3(h; k)与推广的Kloosterman和的混合均值的精确计算公式；给出了带poly-Bernoulli函数的Poly-Dedekind和的定义并证明了其互反公式；给出了带poly-Euler函数的Poly-Dedekind type DC和的定义并证明了其互反公式；研究了Lah–Bell多项式的性质，并得到了其与高阶Bernoulli多项式、poly-Bernoulli多项式等的关系式。所得结果推广了之前的结论，并且为研究一些类Dedekind和或者类Dedekind和与一些特殊函数之间的关系提供理论基础。