关于Dedekind 和的混合均值与互反公式的研究

The main purpose of this project is to study some hybrid mean values of Dedekind sums and its reciprocity formula, and use the main achievements that we obtained to solve some classical problems of number theory. Firstly, we will generalize the problem about hybrid mean value of Dedekind sums to positive integer, or solve them completely. Secondly, we will improve the existing results involving Dedekind sums and its generalized sums, and hope to obtain some sharper asymptotic formulae by combining new weighted arithmetical function. Lastly, we will study the reciprocity formula of general Dedekind sums to general positive integers. Dedekind sums and its related problems play an important role in the study of analytic number theory and combinatorial number theory, and the intersection and fusion of these problems and other branches of mathematics are more and more discovered frequently. Any nontrivial progress in this field will contribute to the development of analytic number theory; even the improvement of fundamental mathematic may be promoted.

本项目主要研究有关Dedekind和的混合均值与互反公式两个方面的问题，同时充分应用所得结果来解决一些经典的数论问题。具体内容：首先，将Dedekind和的混合均值研究中对一般正整数未解决的问题进一步做深入推广或予以彻底解决；其次，改进有关Dedekind和及其推广均值研究的已有的结论，并结合新的加权函数给出新的更好的渐近公式；最后，将广义Dedekind和的互反公式推广到一般的正整数情形。这些内容在解析数论与组合数论研究中占有十分重要的位置， 而且Dedekind和及其相关问题与其它数学分支的交叉融合日益显现，因而在这一领域取得任何实质性的进展，必将对解析数论，甚至对整个基础数学的发展起到重要的推动作用。

本项目主要针对数论中的均值计算与几类重要丢番图方程的可解性这两方面的问题进行研究。一方面，在改进Dedekind和的混合均值计算的基础上，利用解析方法及广义Kloosterman和的性质研究一类特殊的Gauss和的均值估计问题，给出了一个较强的上界估计结果.另一方面，针对椭圆曲线整数点的丢番图方程，指数丢番图方程以及分数丢番图方程等类型的未解决问题做了进一步的深入推广或予以彻底的解决，得到了一些有意义的结论。