Mathematical Sciences: Global Methods in Analytic Number Theory

埃利奥特9530690这个奖项提供资金的调查，将纳入一个驱动器torwards 1904年的迪克森猜想，和著名的素数对问题。这个驱动器将开发利用谐波分析乘法组的有理数。为此，将单模乘法函数看作群上的特征标，系统地研究了它们在大差等差数列上的值分布。除了它们在数论中的内在兴趣和适用性之外，所采用的方法还为用于快速分解大数的各种算法提供了理论方法;因此，这些方法与公共加密有关。方法已经开发的算术函数的研究，暴露在一个AMS的回忆录，更充分地在一本书将出版的剑桥大学出版社，是从根本上重组教学的分析数论在提议者的大学，特别是，适用于培训的几个研究生攻读博士学位。对素模的最小原根的值分布作了辅助研究。该函数与维诺格拉多夫的一个旧猜想有关，并且在计算理论中具有重要意义。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.