Ramanujan tau function

Remarks

Example: the coefficient of $q^2$ comes only from $q \cdot (1 - q)^{24}$, which gives $-24$; so $\tau(2) = -24$. The function is multiplicative: $\tau(mn) = \tau(m)\tau(n)$ when $\gcd(m, n) = 1$. For primes $p$ and $r \ge 1$, $\tau(p^{r+1}) = \tau(p)\tau(p^r) - p^{11} \tau(p^{r-1})$. Non-example: $\tau$ is not completely multiplicative, since $\tau(4) = \tau(2)^2 - 2^{11} = -1472 \neq 576 = \tau(2)^2$. Ramanujan's congruence: $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$, where $\sigma_{11}(n) = \sum_{d \mid n} d^{11}$. For $n = 2$: $\sigma_{11}(2) - \tau(2) = 2049 + 24 = 2073 = 3 \cdot 691$. Whether $\tau(n) \neq 0$ for all $n$ is an open question, known as Lehmer's conjecture.