Dedekind和及其相关问题的研究

Dedekind和是数论中重要的研究领域，也是研究数论问题的重要工具。对它们的研究能促进数论及相关学科的发展，有着重要的理论价值和应用价值。项目申请人在与Dedekind和相关的Bernoulli多项式的研究中有一定的工作基础。因而，本项目计划按照如下方案开展Dedekind和的研究：利用部分分数分解建立任意多个Bernoulli多项式的乘积公式；利用Bernoulli多项式的乘积公式和循环置换研究高维Dedekind-Rademacher和的互反律，并探索函数域上的Dedekind和的类似问题。

Dedekind sums is an important research field in number theory, and is also an important tool to study some problems in number theory. The study on them can promote the development of number theory and related disciplines, has important theoretical value and application value. The applicant has some basic work in the study on the Bernoulli polynomials associated with Dedekind sums. Therefore, this project will be carried out to research Dedekind sums as follows: the formula of products of any number of Bernoulli polynomials will be established by applying the partial fraction decomposition; the reciprocity law of higher dimensional Dedekind-Rademacher sums will be explored in view of using the formula of products of Bernoulli polynomials and the cyclic permutation, and some similar problems for Dedekind sums in function fields will also be considered.