The distribution of prime numbers

Project level: Honours

The Riemann Hypothesis is possibly the most well known unsolved problem in all of mathematics. Incredibly, the more we know about the zeroes of the Riemann zeta-function, the more we know about the set of prime numbers. There are entire books full of unsolved problems on prime numbers; recent advances in studying zeta-function zeroes can be called upon to tackle these.

One such problem involves proving the existence of a function h(x) such that the interval (x, x+h(x)) always contains at least one prime. The goal of this area of research is to find slow-growing functions that do the trick; this would ensure the existence of primes in quite small intervals. Much of this research is motivated by Legendre's conjecture: the unproved assertion that there is always a prime number between any two square numbers.

Another problem involves the additive role that primes play. The Goldbach conjecture is the assertion that any even number greater than two can be expressed as the sum of two primes. There are many related similiar results that one can prove armed with the tools of modern number theory.