Developing an alternative approach to analytic number theory

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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 sums******These projects should push forward the development of the alternative approach.***********************