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，但直到最近才开发出强大的工具来回答世纪的老问题。其中一个主要的工具是复值黎曼zeta函数的素数和零点之间的相互作用。黎曼假设，一个有150年历史的猜想，声称它的所有非平凡零点都位于垂直的1/2线上。我的研究调查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。它也有一个明确的味道，这已经变得越来越相关。例如，陶和赫尔夫戈特有关哥德巴赫猜想的重要工作以及霍夫解决鄂尔多斯覆盖同余猜想的其他工作都依赖于关于素数的明确结果。此外，我的结果可以直接应用于其他领域，如丢番图近似，密码学和计算机科学。我首先强调一系列与估计一些素数计数函数有关的问题。其中一些涉及改善工程分别罗瑟和舍恩菲尔德（素数）和拉马尔和鲁姆利（素数在算术级数）。这两个词仍然经常使用，并被引用了数百次。我最近证明了新的明确的界限psi（x）和psi（x;q，a）通过开发一个优化的平滑方法结合筛选参数，和零自由区域。我计划建立明确的Burgess界限的Dirichlet字符，然后推导出改进的零自由区域的Dirichlet L-函数。对于psi（x），我还使用了zeta的第一个显式零密度估计，对于psi（x;q，a），使用了Platt对广义黎曼假设的广泛数值验证。零密度估计是一个强大的工具，因为我们能够控制零距离1线的距离和数量。我计划进一步调查明确的界限，他们的情况下，黎曼和狄利克雷L-函数。** 我也计划把这些想法中的一些带到数域中的素数研究中。在70年代后期，Lagarias和Odlyzko给出了Chebotarev密度定理的有效版本，然后，与蒙哥马利一起，提供了相关素数计数函数Pi_C（x）的上界，对于更大的x范围。这些定理依赖于Dedekind zeta函数零点的分布。现在有一些明确的结果，关于这些（证明自己或与吴）。我感兴趣的是将这些结果扩展到Hecke L-函数，并将它们应用于改进Ng和我在Chebotarev密度中获得的最小素数。此外，我想证明Serre，Wan和Murty关于Pi_C（x）的一些定理的显式版本。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.**