The way in which disorderly systems like glasses freeze could shed light on one of the greatest enigmas in mathematics today.

The mystery in question concerns prime numbers, which are essentially the elementary particles of arithmetic — a prime number such as 2 is divisible only by 1 and itself, while a composite number such as 4 is divisible by 1, 2 and 4.

One key tool for how prime numbers are distributed in the universe of numbers is the Riemann zeta function. A better understanding how the zeta function works could help mathematicians understand a mysterious pattern in how prime numbers seem to be distributed, upon which many theorems in math rest.


And now scientists are finding remarkable similarities between how disordered systems like glasses freeze and how prime numbers are distributed.

In glasses, atoms are freeze solid, arranged in a disorderly manner, while in crystals, they are arrayed in an orderly fashion. The way energy is distributed within disordered systems like glasses resembles a random landscape of hills and valleys. As the amount of energy within such a system is lowered, any travelers navigating this landscape would slow and eventually stop. The areas in which they would tend to freeze in place resemble the way numbers cluster with the Riemann zeta function.


And that means that a greater understanding of the process of freezing might help tackle one of the greatest unsolved problems in mathematics. The scientists detailed their findings online April 26 in the journal Physical Review Letters.

Image: The zeroes of the Riemann zeta function. Credit: WikiCommons.