Divisibility and PrimesDistribution of the Primes

The Riemann Hypothesis

Mathematicians have spent many centuries exploring the pattern and distribution of prime numbers. They seem to appear completely randomly – sometimes there are huge gaps in between consecutive primes, and sometimes we find twin primes right next to each other.

When only 15 years old, the German mathematician Carl Friedrich Gauss had a groundbreaking new idea: he counted the number of primes up to a certain point, and showed the results in a chart:

Along the x-axis you can see all integers. Whenever there is a prime, the Prime Counting Function (shown in blue) increases by one. As we zoom out, the blue line becomes very smooth. Gauss noticed that the shape of this function looks very similar to the function xlogx (shown in red). He predicted that the two functions are always “approximately similar”, and this was proven in 1896.