Topics in analytic number theory and beyond

在1859年黎曼给出了一个身份的一些素数x方面的零点黎曼zeta函数（这是解析延续的功能是“自然定义”只有一半的复平面）。这个令人信服的身份一直是基础上的研究分布的素数以来和所有的教科书开始从这个角度。事实上，在1949年初等证明的素数定理，它被认为是不可能采取任何不同的方法，甚至在此之后，它似乎是其他方法是“特设”，因此在其适用性有限。 在过去的十年里，Soundararajan和我一直在发展分析数论中的“自命不凡”概念，证明了不同问题的几个开放式命题，并对几个已知定理给出了新的和完全不同的证明。从经典解析数论的角度来看，Linnik定理，即在每个可行的算术级数中存在小素数，是困难和微妙的证明。2009年11月，我们意识到自命不凡的方法如何提供相对容易的证明，导致了“自命不凡的大筛子”和其他新技术的发展。这反过来又使我们认识到，有可能在不使用zeta函数零点的情况下，发展基本经典解析数论的所有结果。如果我们能够成功，这将是自1859年以来对这个非常好的探索主题的第一个新方法！ 我们目前的目标是对达文波特的书中的所有结果，以及（也许）Bubereri的结果，以及各种新的应用，给出自命不凡的证明。

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.