8. How to win a million dollars

Abstract

What is the Riemann hypothesis, and why does it matter? ‘How to win a million dollars’ looks in detail at Riemann’s conjecture. While Gauss attempted to explain why primes thin out, Bernhard Riemann in 1859 proposed an exact formula for the distribution of primes, employing Euler’s ‘zeta function’ and the idea of complex numbers. In 2000, the Clay Mathematics Institute offered a million dollars for the solutions of each of seven famous problems, of which the Riemann hypothesis was one. The Riemann hypothesis implies strong bounds on the growth of other arithmetic functions, in addition to the primes-counting function. It remains one of the most famous unsolved problems of mathematics.