What is the best approach to counting primes?

Abstract

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.