指数和、Dedekind和及其应用研究

关于指数和、Dedekind和的加权均值及其应用研究，一直以来都是数论研究中的重要的研究课题之一。近年来，很多学者对这些问题进行了深入的研究, 并且取得了许多研究成果。然而由于许多数论问题研究的难度非常之大，因此至今所取得的研究进展仍不理想，达不到令人满意的目的，所以对这些问题及其推广形式继续进行深入系统的研究仍是一项很有意义的工作。本项目主要利用解析方法，研究混合指数和及其它相关和的高阶加权均值计算；在建立Dedekind和与广义三项指数和、广义Kloosterman和之间的联系的基础上，研究它们的高次混合均值计算问题，同时，将Dedekind和与二次Gauss和的高次加权均值问题与二次域的类数联系起来，寻找类数的一种新的更加直接的计算方法。在Dedekind和与广义Fibonacci数列的计算问题中，还可计算Dedekind和与其他二阶线性递推数列的计算公式。

The study on the weighted mean value of exponential sums、Dedekind sums and its applications is one of the important issues of the research in number theory. In recent years, many scholars had studied those problems deeply, and obtained many interesting results. However, due to the difficulty of the study of some number theory problems is very large, the research progress achieved so far is still not ideal, amounted to less than satisfactory purpose, so the depth of these issues and their promotion system research is still a meaningful work.This project mainly use the analytic methods to study the high hybrid power mean value calculating problems of exponential sums and other related sums；Based on the relationship between the classical Dedekind sums and general three-term exponential sums、the Kloosterman sums, we study their mean value computational problems. At the same time, we also contact the high power mean value calculating problems of Dedekind sums and quadratic Gauss sums with the class number of quadratic field， to find a new more directly method to calculate that. In computational problems of Dedekind sums and general Fibonacci numbers，we can also calculate computational formula of Dedekind sums and other second-order linear recurrence sequences.