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 \(n\).