Topics in analytic number theory and beyond

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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.