Developing an alternative approach to analytic number theory

1859年，Riemann出版了一本十页的专著，其中他展示了如何通过研究Riemann Zeta函数的零点(出现在其解析连续域中)来实现对素数分布的理解。自那以后，他的非凡方法一直主导着这个主题，导致了我们所知道的关于素数的大部分内容，并允许研究人员穿透关于L的自然分析问题-在许多不同的背景下的函数。然而，这种方法也有令人失望的局限性：在质量上：在过去的50年里，对关键估计的基本改进很少(例如，减少了素数定理中的误差项)；在数量上：似乎没有办法使用这些方法来攻击某些基本问题(例如，证明在[x，x]形式的所有区间中都存在素数)；并且从根本上：这些方法只有当相应的狄里克莱级数可以解析地连续到“临界带”时才适用，但是我们知道如何解析地只连续自然产生于算术中的L函数的有限子集。2009年，Soundararajan和我开始开发一种替代方法来解决这个问题。虽然这在很大程度上是基于早期作者的各种特殊技术，特别是我们所说的自命不凡的概念(从Halasz的结果演变而来)，但当结合到一个逻辑流中时，这为解析数论提供了一个连贯的新视角。事实上，一些经典的技术更适合这个框架。在这个方案中，我们用另一种观点解决了解析数论中的几个著名问题，并发展了新的理论，我们的目的是：--用Koukoulopoulos和Soundarajan在多个方向上推进乘法函数平均值的渐近公式理论；--简化通过三重积积分的理论，扩大范围，获得短区间素数的下界(与Harper，Matomaki和Radziwill)。--得到一个误差项更强(因此更有用)的中值的“结构定理”--得到被乘性函数在主弧上扭曲的指数和的渐近性，从而解决各种三值算术问题，如de la Breteche和Soundararajan。然后，在迈尔森的帮助下，用这项技术解决更高维度的类似问题。--与Myerson一起给出考虑所有形式的自命不凡的一般Barban-Davenport-Halberstam定理(就像我与肖在Bombieri-Vinogradov上的工作一样)--进一步发展加法组合学与格林-陶定理的联系，也许(与Harper一起)给Lipschitz谱一个粗略的分类--更好地理解简短特征总和的分布--这些项目应该会推动替代方法的发展。

In 1859 Riemann published a ten page monograph in which he showed how an understanding of the distribution of prime numbers can be achieved through the study of the zeros of the Riemann zeta function (which occur in its domain of analytic continuation). His extraordinary approach has dominated the subject ever since, leading to most of what we know about primes, and allowing researchers to penetrate natural analytic questions about L-functions in many different settings. However, there are disappointing limitations on this approach: Qualitatively:, There have been few fundamental improvements to key estimates in the last fifty years (eg, reducing the error term in the prime number theorem); Quantitatively: There does not seem to be a way to attack certain fundamental questions using these methods (eg, proving there are primes in all intervals of the form [x, x + x]); and Fundamentally: These methods apply only when the corresponding Dirichlet series can be analytically continued into the “critical strip”, yet we know how to analytically continue only a limited subset of the L-functions that arise naturally in arithmetic. In 2009, Soundararajan and I began developing an alternative approach to the subject. Although largely based on various ad hoc techniques of earlier authors, particularly what we call the notion of pretentiousness (evolved from a result of Halasz), when combined into one logical flow, this presents a coherent new perspective on analytic number theory. Indeed some of the classical techniques fit better into this framework. In this proposal we attack several well-known questions in analytic number theory using the alternative perspective and also develop the new theory, We aim to: -- Push forward the theory of asymptotic formulas for mean values of multiplicative functions, in several directions, with Koukoulopoulos and Soundararajan; -- Simplify the theory of integrating via triple products, improve the range and obtain lower bounds for the number of primes in short intervals (with Harper, Matomaki and Radziwill). -- Obtain a "structure theorem" for mean values with a stronger (and so more useful) error term -- Get asymptotics for exponential sums twisted by multiplicative functions, on the major arcs, so as to solve various ternary arithmetic problems, with de la Breteche and Soundararajan. Then, with Myerson, use this technology for analogous questions in higher dimensions. -- Give, with Myerson, a general Barban-Davenport-Halberstam theorem taking account of all forms of pretentiousness (as in my work with Xiao on Bombieri-Vinogradov) -- Develop further links for additive combinatorics with a route to the Green-Tao theorem, and perhaps give a rough classification to the Lipschitz spectrum (with Harper) -- Better appreciate the distribution of short character sumsThese projects should push forward the development of the alternative approach.