Zeros of L functions and distribution of primes

虽然素数的研究是一个古老的研究领域，但直到最近才有强大的工具被开发出来，为一个世纪以来的问题带来答案。主要工具之一是复值Riemann Zeta函数的素数和零点之间的相互作用。有150年历史的黎曼假设断言，它所有的非平凡零点都位于垂直的1/2直线上。研究了Zeta函数(和其他L函数)的零点的位置和密度以及素数的分布。它也有一种明显的味道，这已经变得越来越相关。例如，陶和赫夫戈特关于Goldbach猜想的重要工作，以及Hough解决Erdos覆盖同余猜想的其他工作，都依赖于关于素数的显式结果。此外，我的结果还可以直接应用于其他领域，如丢番图逼近、密码学和计算机科学。

While the study of prime numbers is an antique field of study, it is only recently that powerful tools have been developed to bring answers to century old questions. One of the main tools is the interplay between the primes and the zeros of the complex valued Riemann zeta function. The Riemann Hypothesis, a 150 year old conjecture, asserts that all its nontrivial zeros lie on the vertical 1/2-line. My research investigates the location and density of the zeros of the zeta function (and other L-functions) and the distribution of primes. It also has an explicit flavor which has become increasingly relevant. For instance, important work by Tao and Helfgott related to Goldbach's conjecture and other work by Hough solving Erdos covering congruences conjecture rely on explicit results about the primes. Moreover my results can be directly applied in other fields, such as Diophantine approximation, cryptography, and computer science.