Distribution of Primes and Other Topics in Analytic Number Theory

素数在各种整数序列中的分布在数论中具有极其重要的意义。这种分布的研究同时使用乘法和加性数论。使用的一些工具包括圆法，指数和，筛子和zeta函数方法。本项目将继续早期的工作，以获得各种序列中素数对的下界。该方法使用短除数和，并与上述所有工具都有联系。研究者将进一步发展和推广这种方法。他还将研究质数在等差数列和质数多重相关中的应用。一个相关的主题是当平均可用时获得渐近结果。此外，研究者将研究涉及幂和法的一些问题；这些问题很适合本科院校。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。

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.