Multiplicative Number Theory and L-functions

这个项目的第一个主要目标是进一步加深我们对整数，特别是连续整数分解方式的理解。这是一个基本的数学问题，在密码学(例如RSA)和数学物理(例如研究一般环面上粒子能级之间的间隙)等领域都有广泛的应用。这个项目的第二个主要目标是应用在这个问题上的进展来进一步发展纯数学的某些领域，特别是L函数的分析理论。该项目包括对研究生和博士后的培训。为了达到第一个目标，PI的目标是在Chowla猜想，特别是“乘法函数的局部傅立叶一致猜想”方面取得进展。后者的完全解析将意味着Chowla对于任意数目的移位的对数形式的猜想，特别是以对数形式的Sarnak猜想。对于第二个目标，PI旨在建立在L函数分析理论和乘数理论之间的类比基础上，以在与L函数的非零性、值分布和矩相关的问题上取得进展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The first main objective of this project is to develop further our understanding of the way in which integers and especially consecutive integers factorize. This is a fundamental mathematical problem which finds wide ranging application in fields as distant as cryptography (e.g RSA) and mathematical physics (e.g the study of gaps between energy levels of particles on generic tori). The second main objective of this project is to apply the progress on this question to further develop certain areas of pure mathematics, in particular the analytic theory of L-functions. The project includes training of graduate students and postdocs.Towards the first goal the PI aims to make progress on Chowla's conjecture and in particular on the ``local Fourier uniformity conjecture for multiplicative functions''. A full resolution of the latter will imply Chowla's conjecture in logarithmic form for an arbitrary number of shifts and in particular Sarnak's conjecture in logarithmic form. Towards the second goal the PI aims to build on analogies between the analytic theory of L-functions and multiplicative number theory to make progress on questions related to the non-vanishing, value distribution and moments of L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.