一类带有算术函数指数和的几个应用

项目名称： 一类带有算术函数指数和的几个应用

项目编号： No.11301325

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 姚维利

作者单位： 上海大学

项目金额： 22万元

中文摘要： 获得指数和较为精确的估计是十分重要的,解析数论及密码学中许多重要问题的处理与之相关。尽管如此，这方面的研究远不够完善和系统，仍有许多问题值得研究和探讨。 本项目通过对Huxley-Hooley围道进行修改，综合利用复积分法及指数和方法，系统地研究一类带有算术函数的指数和，建立其Bombieri 型定理。进而，采用不同于传统的方法，探讨此结果的几个应用： 一、研究华林问题。结合圆法及解析方法，可无条件地扩张主区间，并能很好地处理扩张后的主区间；二、结合大筛法不等式、零点密度估计等，研究级数中具有固定个素因子的整数之分布。获得其Barban-Davenport-Halberstam型定理，提高经典结论的误差；综合利用圆法，可进一步获得其较为精确的渐近公式，推广素数分布的相关结果；三、为丢番图问题的研究提供必要的理论支持和方法指导，延伸和推广前人的经典之作。

中文关键词： 指数和；Huxley-Hooley 围道；零点密度；圆法；Bombieri型定理

英文摘要： It is really important to obtain sharper estimates for exponential sums, since they are closely related to the research of many important problems from analytic number theory and cryptography. However, studies in this field are far from perfect and systematic. There are still lots of problems which are worth researching. This project presents a systematic study on a kind of exponential sums involving arithmetic functions, and gives a Bombieri-type theorem for it, by modifying the Huxley-Hooley contour and applying complex integral methods. The main purpose of this project is to study several applications of the theorem (with various forms) to the following problems. More precisely: First, the Waring problems. Combining our theorem with the circle method and anaytic methods, we can unconditionally enlarge the major arcs, and then handle the enlarged arcs. Second, the distribution of integers with fixed number of primes in progressions. Applying our theorem, among other things, and large sieve inequality, as well as zero density estimates, we can give a Barban-Davenport-Halbersta type theorem for this distribution, and sharpen the error term of the previous results. Furthermore, we also can generalize the results of prime distribution by giving a stronger asymptotic formula. Finally, the Diophanti

英文关键词： exponential sums；Huxley-Hooley contour；zero-density；circle method；Bombieri-type mean value