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。*我首先强调一些与估计一些素数计数函数有关的问题。其中一些涉及到罗瑟和舍恩菲尔德(对于素数)和拉马雷和鲁姆利(对于算术级数中的素数)作品的改进。这两个词仍然被频繁使用，并被引用了数百次。我最近证明了psi(X)和psi(x；q，a)的新的显式界，方法是开发了一种优化的平滑方法，并结合筛选变元和无零区域。我计划建立Dirichlet特征标的显式Burgess界，然后推导出Dirichlet L-函数的改进的无零域。对于psi(X)，我也使用了zeta的第一个显式零密度估计，对于psi(x；q，a)，我使用了Platt对广义黎曼假设的广泛的数值验证。零密度估计是一个强大的工具，因为我们能够控制零点离1线的距离有多远，零点有多少。我计划在黎曼函数和狄里克莱特L函数的情况下进一步研究它们的显界。*我还计划把这些想法中的一些带到数域中素数的研究中。七十年代末，Lagarias和Odlyzko给出了Chebotarev密度定理的有效形式，然后与Montgomery一起给出了较大范围内素数计数函数PI_C(X)的上界，这些定理依赖于DedekindZeta函数的零点的分布。关于这些，现在已经有了一些明确的结果(由我自己或与Ng共同证明)。我感兴趣的是将这些结果推广到Hecke L-函数，并应用它们来改进我和Ng在Chebotarev密度下得到的关于最小素数的界。此外，我还想证明关于PI_C(X)的Serre，wan和Murty的一些定理的显式形式。最后，我对一些应用感兴趣，包括Lang-Trotter猜想的素数计数函数的界。

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.******I first highlight a list of problems related to estimating some prime counting functions. Some of them concern improvements of works of respectively Rosser and Schoenfeld (for the primes) and of Ramare and Rumely (for the primes in arithmetic progressions). These are both still used frequently and referenced hundred of times. I recently proved new explicit bounds for psi(x) and psi(x;q,a) by developing an optimized smoothing method in conjunction with a sieving argument, and zero-free regions. I plan to establish explicit Burgess bounds for Dirichlet characters and to then deduce improved zero-free regions for Dirichlet L-functions. For psi(x), I also used the first explicit zero density estimate for zeta, and for psi(x;q,a), the extensive numerical verifications of Platt for the Generalized Riemann Hypothesis. Zero density estimates are a powerful tool as we are able to control how far and how few the zeros are from the 1-line. I plan to investigate further explicit bounds for them in the case of both the Riemann and Dirichlet L-functions. ******I also plan to bring some of these ideas in the study of primes in number fields. In the late seventies, Lagarias and Odlyzko gave effective versions of the Chebotarev Density Theorem and then, with Montgomery, provided upper bounds for the associated prime counting function Pi_C(x) for a larger range of x. These theorems relied on the distribution of the zeros of Dedekind zeta functions. There are now some explicit results concerning these (proven by myself or jointly with Ng). I am interested in extending these results to Hecke L-functions and to apply them to improve the bound Ng and I obtained for the least prime in the Chebotarev density. Moreover, I would like to prove explicit versions of some theorems of Serre, Wan, and Murty on Pi_C(x). Finally, I am interested in some applications including bounds for the prime counting functions of the Lang-Trotter conjecture.**