Mathematical Sciences: Global Methods in Analytic Number Theory

埃利奥特9530690这个奖项为一项调查提供了资金，该调查将包括1904年迪克森猜想和著名的素数对问题。这种传动将利用有理数乘法群上的调和分析来开发。为此，把么模乘法函数看作群上的特征标，系统地研究了它们在差异较大的算术级数上的值分布。除了它们在数论中的内在兴趣和适用性之外，将要使用的方法还为用于大数的快速因式分解的各种算法提供了理论上的方法；因此，这些方法与公共加密有关。在剑桥大学出版社即将出版的一本书中，部分地暴露在AMS回忆录中的算术函数的研究方法将被用来从根本上改革提出者所在大学的解析数论的教学，特别是应用于几个攻读博士学位的研究生的培训。一个素数模的最小本原根的值分布将被辅助研究。该函数与维诺格拉多夫的一个古老猜想有关，在计算理论中具有重要意义。这项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，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。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领域的各种应用中不可或缺的工具。

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.