算术函数的均值及相消性研究

算术函数是定义在正整数集合上取值为复数的函数，是数论研究中关键而基本的研究对象，很多著名而重要的数论问题都与之密切相关，最终都可归结或转化为某种算术函数的上下界估计或某种形式的均值估计问题，其本质是这些算术函数的值分布问题。该项目拟研究Dirichlet特征、特征和、多项式指数和等算术函数在不同集合、不同区间上、不同加权下的值分布规律或抵消现象以及它们的不同形式的均值的上下界估计或渐近性质，研究Dedekind和、Hardy和、广义Dedekind和、m次Dedekind和、Cochrane和等Dedekind型和以及Lehmer问题及其推广问题的误差函数等几类算术函数的均值分布与相消性，并结合以上几类算术函数的值分布规律从一般意义上研究算术函数值的抵消与共振现象。

Arithmetic functions are complex value functions defined on positive integers. They are important and basic research objects in number theory and have colse relations with many number theory problems. The key points of this problems are the upper or lower bounds or some kind of mean value of some arithmetic functions. We will study the mean value distribution, upper or lower bound, and cancellations of some some arithmetic functions such as Dirichlet characters, character sums, polynomial exponential sums, Dedekind sums, Hardy sums, general Dedekind sums, m-th Dedekind sums,error functions of D.H.Lehmer problem and its generalization over different sets, intervals and with different weight. Combining the value distribution law of these arithmetic functions, we will also study the cancellations and resonance of any arithmetic functions from general points.