积性函数的均值及其应用

Mean value of multiplicative functions is one of the key issues in the study of analytic number theory. It has many connections with the unsolved problems such as Riemann Hypothesis. In recent years, A. Granville and K. Soundararajan systematically developed the general theory of multiplicative functions and obtained many deep results. On the other hand, due to the introducing of ideas from dynamical systems and Fourier analysis by P. Sarnak, N. Frantzikinakis and B. Host, the study of many classical problems for mean value of multiplicative functions is pushed ahead dramatically. The main purpose of the present project is to study some important variants of Chowla conjecture, multiplicative function approach to character sums over shifted primes, and the general theory of multiplicative functions in function fields with various potential applications in number theory. Substantive progress is expected in the above topics.

积性函数的均值一直以来都是解析数论研究的关键问题，并与 Riemann 假设等未解决的问题有着紧密的联系。近年来，A. Granville 与 K. Soundararajan 等人系统地发展了关于积性函数的一般理论，获得了深刻的结果。另一方面，由于 P. Sarnak, N. Frantzikinakis 及 B. Host 等人将动力系统及 Fourier 分析的思想引入其中，使得积性函数均值的很多经典问题获得了新的推动力。本项目主要研究二维 Chowla 猜想的重要变体，平移素变数特征和的积性函数新途径，函数域上积性函数均值的一般理论及其在经典数论问题中的应用，期待得到实质性的进展。

本项目对积性函数均值及相关问题展开研究,获得了一些重要进展.具体地,我们(1)给出带有积性系数的平移特征和的非平凡上界估计,所得结果具有相当的深度与一般性;(2)对不同模的二次剩余、非剩余、原根在平移素数序列中的联立分布问题给出了完整的解决;(3)给出n维格逼近定理在现有方法框架下的最佳可能误差估计;(4)给出双曲素格点上Dirichlet特征和的非平凡上界估计;(5)给出一类特殊算术序列中最小k次非剩余的显式上界估计;(6)给出平坦数上Legendre符号和上界估计的一个简短而漂亮的证明;(7)证明Lehmer型同余关系式.