Ramanujan's tau function

Ramanujan's tau function, denoted as τ(n), is a special mathematical function that assigns a number to each positive integer n, revealing deep properties in number theory. It arises from the Fourier coefficients of the modular form called the cusp form Δ(z). These coefficients encode complex patterns related to prime numbers and have remarkable multiplicative properties, meaning the value at a product often relates to the values at its factors. The tau function is significant for its connections to prime distribution, factoring, and various areas of mathematics, showcasing the hidden order and symmetry within the integers.