Ramanujan tau function

Summary

The Ramanujan tau function $\tau(n)$ gives the Fourier coefficients of the cusp form $\Delta = q \prod_{n \ge 1} (1 - q^n)^{24}$ of weight $12$. Ramanujan studied it in 1916 and conjectured that it is multiplicative, which Mordell proved in 1917. Its properties come from the theory of Hecke operators. The bound $\abs{\tau(p)} \le 2 p^{11/2}$ for primes $p$ is the Ramanujan–Petersson conjecture, proved by Deligne.