Topics in analytic number theory and beyond

1859年，黎曼用黎曼Zeta函数的零点给出了素数到x的个数的一个恒等式(Riemann Zeta函数是一个函数的解析延拓，该函数只在复平面的一半上“自然定义”)。从那时起，这个令人信服的身份一直是研究素数分布的基础，所有的教科书都是从这个角度出发的。事实上，在1949年素数定理的初等证明之前，人们认为不可能采取任何不同的方法，即使在那之后，似乎其他方法也是“临时的”，因此它们的适用性有限。 在过去的十年里，Soundararajan和我一直在发展解析数论中的“自命性”的概念，证明了关于不同问题的几个公开猜想，并给出了几个已知定理的新的和完全不同的证明。从经典解析数论的角度来看，林尼克定理，即每一个可行的算术级数中都有小素数，证明起来既困难又微妙。2009年11月，我们意识到自命不凡的方法给出了一个相对容易的证明，导致了“自命不凡的大筛子”和其他新技术的发展。这反过来又使我们意识到，有可能发展基本经典解析数论的所有结果，而不需要使用Zeta函数的零点。如果我们能够成功，这将是自1859年以来对这一非常深入探索的主题的第一个新方法！ 我们目前的目标是为达文波特的书中的所有结果，以及(也许)庞比耶里的结果，以及各种相当新的应用提供虚假的证据。

In 1859 Riemann gave an identity for the number of prime numbers up to x in terms of the zeros of the Riemann zeta-function (which is the analytic continuation of a function that is "naturally defined" only on half of the complex plane). This compelling identity has been the basis of the study of the distribution of prime numbers ever since and all textbooks start from this perspective. Indeed before the 1949 elementary proof of the prime number theorem it was believed to be impossible to take any different approach and even after that it had seemed that other approaches are "ad hoc", and so limited in their applicability. Over the last decade, Soundararajan and I have been developing the notion of "pretentiousness" in analytic number theory, proving several open conjectures on different problems, and giving new and quite different proofs of several known theorems. From the perspective of classical analytic number theory, Linnik's Theorem, that there are small primes in every feasible arithmetic progression, is difficult and subtle to prove. In November 2009, we realized how pretentious methods give a relatively easy proof, leading to the development of the "pretentious large sieve" and other new techniques. This in turn has led us to realize that it may be possible to develop all of the results of basic classical analytic number theory, without ever using the zeros of zeta functions. If we can succeed this will give the first new approach to this very well explored subject since 1859! Our current goal is to give pretentious proofs of all of the results in Davenport's book, as well as (perhaps) that of Bombieri, and various quite new applications.