Developing an alternative approach to analytic number theory

1859年，黎曼发表了一篇十页的专著，他在其中展示了如何通过研究黎曼ζ函数的零点（零点出现在黎曼ζ函数的解析延拓域中）来理解素数的分布。从那以后，他的非凡方法一直主导着这个学科，导致了我们对素数的大部分了解，并使研究人员能够在许多不同的环境中深入了解关于l函数的自然分析问题。然而，这种方法有一些令人失望的局限性：定性地说，在过去的50年里，对关键估计几乎没有根本性的改进（例如，减少了素数定理中的误差项）；定量：似乎没有办法使用这些方法来解决某些基本问题（例如，证明在[x， x + x]形式的所有间隔中都存在素数）；这些方法仅适用于相应的狄利克雷级数可以解析连到“临界带”时，然而我们知道如何解析连到算术中自然出现的l函数的有限子集。2009年，Soundararajan和我开始研究这个问题的另一种方法。虽然很大程度上基于早期作者的各种特殊技术，特别是我们称之为自命不凡的概念（从Halasz的结果演变而来），当结合成一个逻辑流时，这提出了分析数论的一个连贯的新视角。事实上，一些经典技术更适合这个框架。在这个建议中，我们使用替代的观点来攻击解析数论中几个众所周知的问题，并发展新的理论，我们的目标是：——与Koukoulopoulos和Soundararajan一起，在几个方向上推进乘法函数均值的渐近公式理论；——简化了三重积积分理论，改进了短区间内素数的取值范围，得到了素数的下界（与Harper， Matomaki和Radziwill合著）。-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --然后，与Myerson一起，将该技术用于更高维度的类似问题。——与迈尔森一起，给出一个考虑了所有形式的伪称的一般Barban-Davenport-Halberstam定理（就像我与肖在Bombieri-Vinogradov上的工作一样）——与格林-陶定理（Green-Tao theorem）的路线进一步发展加性组合学的联系，也许给利普希茨谱（Lipschitz spectrum）一个粗略的分类（与哈珀一起）——更好地理解短字符和的分布这些项目应该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.