# Ramanujan function

The function $n \mapsto \tau(n)$, where $\tau(n)$ is the coefficient of $x^n$ ($n \ge 1$) in the expansion of the product $$D(x) = x \prod_{m=1}^\infty (1 - x^m)^{24}$$ as a power series: $$D(x) = \sum_{n=1}^\infty \tau(n) x^n \ .$$ If one puts $$\Delta(z) = D(\exp(2\pi i z))$$ then the Ramanujan function is the $n$-th Fourier coefficient of the cusp form $\Delta(z)$, which was first investigated by S. Ramanujan . Certain values of the Ramanujan function: $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$, $\tau(6) = -6048$, $\tau(7) = -16744$, $\tau(30) = 9458784518400$. Ramanujan conjectured (and L.J. Mordell proved) the following properties of the Ramanujan function: it is a multiplicative arithmetic function $$\tau(mn) = \tau(m) \tau(n) \ \text{if}\ (m,n) = 1 \,;$$ and $$\tau(p^{n+1}) = \tau(p^n)\tau(p) - p^{11} \tau(p^{n-1}) \ .$$
Consequently, the calculation of $\tau(n)$ reduces to calculating $\tau(p)$ when $p$ is prime. It is known that $|\tau(p)| \le p^{11/2}$ (see Ramanujan hypothesis). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence $$\tau(p) \equiv 1 + p^{11} \pmod{691} \ .$$
Examples of congruence relations discovered later are: $$\tau(n) \equiv \sigma_{11}(n) \pmod{2^{11}} \ \text{if}\ n \equiv 1 \pmod 8$$ $$\tau(p) \equiv p + p^{10} \pmod{25}$$ etc.