Interactions between L-functions and sieve methods

最近解析数论正在经历一场革命。许多老问题已经得到解决，而且技术的大量综合和简化正在发生。首先，在$L$-函数的理论中，我们现在理解了$L$-函数的更精细的行为是由随机矩阵统计决定的。许多工作都是为了验证这些假设，无论是无条件的还是有条件的假设黎曼假设。在这里，哈珀最近在黎曼假设的条件下获得了关于矩的惊人结果，导致了矩问题的近似解。第二，在筛子理论中，戈德斯通-平茨-耶尔德林的方法被认为比预期的要强大得多。例如，它导致了一个相对初级的证明存在有界差距之间的素数，由于梅纳德道。这给了一个不同的方法，以前的深入工作张 *。第三，在自守形式的研究中，量子唯一遍历性的中心问题已经由Holowinsky和Soundararajan以及Lindenstrauss* 解决，我建议研究来自筛法理论的方法与通常用于L-函数研究的方法之间的相互联系。这将不时地依赖于从概率中引进技术，这些技术目前在这两个学科中都很发达，并且可能也会在自守形式的研究中得到副作用。在我的工作筛的长期目标是使1）进一步的进展Chowla的猜想，2）调查，如果类似的结果可以获得素数和3）调查Sarnak的猜想熵的刘维函数。其中一些项目特别是2）可能在密码学方面有值得注意的应用，例如，它们可能导致证明存在无穷多个素数p，使得p-1有一个大的素因子。在我的工作L-函数的长期目标是取得进一步的进展，对austertures的基廷-斯奈思的统计行为的中心值的L-函数，并获得更好的理解最近的方法Aistleitner和Bondarenko-Seip允许一个工作，非常长，但稀疏狄利克雷多项式。

Recently analytic number theory is undergoing a revolution. Many old questions have been solved and moreover a vast synthesis and simplification of the techniques is taking place.******First in the theory of $L$-functions, we now understand conjecturally that the finer behavior of an $L$-function is governed by random matrix statistics. Much work went into verifying these conjectures, either unconditionally or conditionally on the assumption of the Riemann Hypothesis. Here spectacular results on moments have been recently obtained by Harper conditionally on the Riemann Hypothesis, leading to a near solution of the moment problem.******Second, in sieve theory the method of Goldston-Pintz-Yildrim has been understood to be much more powerful than expected. For example it led to a relatively elementary proof of the existence of bounded gaps between primes, due to Maynard-Tao. This gave a different approach to the previous deep work of Zhang.******Third, in the study of automorphic forms, the central problem of Quantum Unique Ergodicity has been resolved by Holowinsky and Soundararajan and by Lindenstrauss******I propose to investigate the inter-connection between methods coming from sieve theory and those usually used in the study of L-functions. This will rely from time to time on importing techniques from probability, which are currently flourishing in both subjects, and will likely have also side applications to the study of automorphic forms.******In my work on the sieve the long term goal is to make 1) further progress on Chowla's conjecture, 2) investigate if analogous results could be obtained for primes and 3) investigate Sarnak's conjecture on the entropy of the Liouville function. Some of these projects in particular 2) could have noteworthy applications for cryptography, for example they could lead to showing that there are infinitely many primes p such that p-1 has a large prime factor.******In my work on L-functions the long term goal is to make further progress on the conjectures of Keating-Snaith on the statistical behavior of central values of L-functions and to gain a better understanding of the very recent methods of Aistleitner and Bondarenko-Seip allowing one to work with very long but sparse Dirichlet polynomials.