Multiplicative Number Theory and L-functions

这个项目的第一个主要目标是进一步发展我们对整数，特别是连续整数的分解方式的理解。这是一个基本的数学问题，广泛应用于密码学（如RSA）和数学物理（如研究一般环面上粒子能级之间的间隙）等领域。这个项目的第二个主要目标是应用这个问题的进展来进一步发展纯数学的某些领域，特别是l函数的解析理论。该项目包括培养研究生和博士后。为了实现第一个目标，PI的目标是在Chowla猜想上取得进展，特别是在“乘法函数的局部傅立叶均匀性猜想”上。后者的完全解决将意味着Chowla猜想对任意移位数的对数形式，特别是Sarnak猜想的对数形式。对于第二个目标，PI旨在建立l函数的解析理论和乘法数论之间的类比，以在与l函数的非消失，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知识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.