Interactions between L-functions and sieve methods

点击翻译按钮获取中文摘要

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.