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，我们现在推测地理解L函数的更精细的行为是由随机矩阵统计控制的。很多工作都是为了无条件地或有条件地根据黎曼假设来验证这些猜想。在这里，Harper最近在黎曼假设上有条件地得到了关于矩的惊人结果，从而导致了矩问题的近似解。******第二，在筛理论中，Goldston-Pintz-Yildrim方法被认为比预期的要强大得多。例如，由于梅纳德-道，它导致了素数之间存在有界间隙的相对初级的证明。这为张之前的深入研究提供了一种不同的途径。******第三，在自同构形式的研究中，Holowinsky和Soundararajan以及Lindenstrauss已经解决了量子唯一遍历性的中心问题******我建议研究筛理论和l函数研究中常用的方法之间的相互联系。这将不时地依赖于从概率论中引入技术，这些技术目前在这两个学科中都很盛行，并且可能也会在自同构形式的研究中有侧面应用。******在我对筛子的研究中，我的长期目标是：1)在Chowla猜想上取得进一步的进展，2)研究是否可以在素数上得到类似的结果，3)研究Sarnak关于Liouville函数熵的猜想。其中一些项目（特别是2）可能对密码学有重要的应用，例如，它们可能会导致证明有无限多个素数，使得p-1有一个大的素数因子。******在我对l -函数的研究中，长期目标是在Keating-Snaith关于l -函数中心值的统计行为的猜想上取得进一步进展，并更好地理解aisstleitner和bondarenco - seip的最新方法，这些方法允许人们使用很长但稀疏的狄利克雷多项式。

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.