Mathematical Sciences: Global Methods in Analytic Number Theory

这个奖项为一项研究提供了资金，该研究将包括对1904年迪克森猜想的推动，以及著名的质数对问题。这种驱动将利用有理数的乘法群上的谐波分析来发展。为此，将单模乘法函数看作群上的字符，系统地研究了它们在差较大的等差数列上的值分布。除了它们在数论中固有的兴趣和适用性外，所采用的方法为大数快速分解的各种算法提供了理论途径；因此，这些方法与公共加密有关。方法已经为算术函数的研究开发,部分暴露在AMS回忆录,更充分地在一本由剑桥大学出版社出版,被用来彻底重组教学解析数论的申请人的大学,特别是应用于训练的几个研究生朝着一个博士一个辅助研究是最原始的根值分布的一个典型的模量。这个函数与维诺格拉多夫的一个老猜想有关，在计算理论中具有重要的意义。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

Elliot 9530690 This award provides funding for an investigation that will incorporate a drive torwards a 1904 conjecture of Dickson, and the celebrated prime-pair problem. This drive will be developed using harmonic analysis on the multiplicative group of the rationals. To this end, unimodular multiplicative functions are to be viewed as characters on the group, and a systematic study made of their value distribution on arithmetic progressions with large differences. Besides their intrinsic interest and applicability in number theory, the methods to be employed offer a theoretical approach to various algorithms used in the fast factorization of large numbers; hence, these methods bear on public encryption. Methods already developed for the study of arithmetic functions, exposed in part in an AMS Memoir, more fully in a book to be published by Cambridge University Press, are to be employed to radically restructure the teaching of Analytic Number Theory in the proposer's university, and, in particular, applied to the training of several graduate students working towards a Ph. D. An auxiliary study is to be made of the value distribution of the least primitive root of a prime modulus. The function is connnected to an old conjecture of Vinogradov, and of vital interest in the theory of computing. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.