Distribution of Primes and Other Topics in Analytic Number Theory

Goldston 9626903素数在各种整数序列中的分布在数论中是至关重要的。对这种分布的研究使用了乘法和加法两种数论。使用的一些工具包括圆法、指数和法、筛法和Zeta函数法。这个项目将继续早期关于获得各种序列中的素数对的下界的工作。该方法使用较短的除数和，并与上述所有工具相关联。研究人员将进一步发展和推广这一方法。他还将研究素数在算术级数和素数的多重相关性中的应用。一个相关的主题是当平均化可用时获得渐近结果。此外，研究人员将研究涉及幂和方法的某些问题；这些问题非常适合本科生院校。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。

Goldston 9626903 The distribution of primes in various sequences of integers is of critical importance in Number Theory. The study of this distribution uses both multiplicative and additive number theory. Some of the tools utilized include the circle method, exponential sums, sieves, and the zeta function methods. This project will continue earlier work on obtaining lower bounds for pairs of primes in various sequences. The method uses short divisor sums and has connections with all of the aforementioned tools. The investigator will further develop and generalize this method. He will also examine applications to primes in arithmetic progressions and multiple correlations of primes. A related topic is to obtain asymptotic results when averaging is available. In addition, the investigator will study certain problems involving the power sum method; these problems are well suited to an undergraduate institution. 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.