Entry 22

The Ramanujan tau function $\tau(n)$ is given by the coefficients of the q-expansion of the Dedekind eta function $\eta(z)$'s $24$th power.  Let $q = e^{2\pi i\,z}$, then $$\begin{aligned}\eta(z)^{24} &= \sum_{n=1}^\infty\tau(n)q^n\\&=q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - 6048q^6 - 16744q^7 + \dots\end{aligned}$$ Let $n$ be a prime $p$. Ramanujan observed the remarkable congruence $$\tau(p)-1-p^{11}\equiv 0\ \bmod\ 691$$ For example $$\begin{aligned}-24-1-2^{11}&= -691\times3\\252-1-3^{11}&= -691\times256\\4830-1-5^{11}&= -691\times70656\\-16744-1-7^{11}&= -691\times2861568\end{aligned}$$ and so on. More generally, what he observed was $$\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691$$ where $\sigma_k(n)$ is the sum of the $k$th powers of the divisors of $n$.