Mathematical Sciences: Multiplicative and Additive Theory ofPrime Numbers and Related Topics in Analytic Number Theory

本研究涉及素数理论，素数理论可分为乘法和加法两种方法。乘法理论主要涉及Riemann Zeta函数的素数和零点之间的对偶性。加法理论基于素数的指数和，通常使用特征标和狄里克莱特L函数来将加法特征标表示为乘法函数的和。这两个领域之间的重叠是本研究要研究的问题。要考察的一些具体课题包括Zeta函数零点的对相关性，素数对及其与Klostermann和和Barban-Davenpoort型定理的联系，Zeta零点之间的小间隙，函数S(T)和西格尔零点。这项研究涉及到素数理论，即素数在整数中的分布。某些复杂的分析函数，称为Zeta函数，包含这些信息，因此使用分析中的工具来研究它们。因此，这门课是数论和分析的完美结合。结果非常深刻，来之不易。戈德斯顿在解决这一领域的问题方面取得了显著的成功。这项研究肯定会极大地推动我们在这个重要的数学领域的知识。

This research concerns prime number theory, which can be divided into two approaches, the multiplicative and the additive. The multiplicative theory is largely concerned with the duality between primes and zeros of the Riemann zeta function. The additive theory is based on exponential sums of primes and typically uses characters and Dirichlet L-functions in order to express additive characters as sums of multiplicative functions. The overlap between these two areas is what is being studied in this research. Some specific topics to be examined are pair correlation of zeros of zeta functions, prime pairs and their connection with Kloostermann sums and Barban-Davenpoort type theorems, small gaps between zeta zeros, the function S(T) and Siegal zeros. This research concerns prime number theory, that is, the distribution of the prime numbers in the integers. Certain complex analytic functions, called zeta functions, contain this information and so they are studied uses tools from analysis. Thus this subject is a nice blending of number theory and analysis. The results are very deep and difficult to come by. Goldston has been remarkably successful in solving problems in this area. This research will certainly push forward our knowledge substantially in this important area on mathematics.