Ramanujan's hypothesis

The tau conjecture, also known as Ramanujan's hypothesis after its proposer, states that τ(n)~O(n^(11/2 + ϵ)), where τ(n) is the tau function. This was proven by Deligne in the course of proving the more general Petersson conjecture. Deligne was awarded the Fields medal for his proof.

Petersson conjecture | tau function