Abstract

Since Riemann’s 1859 monograph, the study of the distribution of prime numbers (and other arithmetic objects) has been dominated by the study of the zeros of the Riemann zeta function and Dirichlet L-functions (and other, higher dimensional, L-functions). Although there have been ad hoc elementary proofs of some of the key results, there has been no coherent alternative approach to that of Riemann. In this talk we will motivate and outline a new and different way to develop analytic number theory, without zeros, stemming from the concept of “pretentiousness”. This is jointly developed with K. Soundararajan, and we will discuss further developments due to Dimitris Koukoulopoulos, Adam Harper, and several others.