If a solution to the Riemann hypothesis is confirmed, it would be big news. Among other things, the hypothesis is intimately connected to the distribution of prime numbers, those indivisible by any whole number other than themselves and one. If the hypothesis is proven to be correct, mathematicians would be armed with a map to the location of all such prime numbers, a breakthrough with far-reaching repercussions in the field.
Before I posted this, I didn’t know anything about the Riemann hypothesis so I did some digging. The Wikipedia article is full of technical terminology so if you’re a layman like me, this’ll make more sense (from the Simple English version of Wikipedia):
The hypothesis is named after Bernhard Riemann. It is about a special function, the Riemann zeta function. This function inputs and outputs complex number values. The inputs that give the output zero are called zeros of the zeta function. Many zeros have been found. The “obvious” ones to find are the negative even integers. This follows from Riemann’s functional equation. More have been computed and have real part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2.
The functional equation also says all zeros (except the “obvious” ones) must be in the critical strip: real part is between 0 and 1. The Riemann hypothesis says more: they are on the line given, in the image on the right (the white dots). If the hypothesis is false, this would mean that there are white dots which are not on the line given.
If proven correct, this would allow mathematicians to better describe how the prime numbers are placed among whole numbers.
Short version: it deals with prime numbers and if Atiyah has a proof, it’ll make prime number discovery and understanding much easier.
Atiyah has a paper on this topic, too, if you’re really into math.