TITLE: Ramanujan's tau function
QUESTION [16 upvotes]: Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I know the function's use and value to Ramanujan at the time.

REPLY [11 votes]: Why not simply looking at the original source?
Ramanujan made his famous conjectures in On certain arithmetical functions Transactions of the Cambridge Philosophical Society XXII (1916), a source which is easily available (here for instance). In this paper, he intends to study the sum
\begin{equation}
\sum_{r,s}(n)=\sum_{i=0}^{n}\sigma_{r}(i)\sigma_{s}(n-i)=\sigma_{r}(0)\sigma_{s}(n)+\cdots+\sigma_{r}(n)\sigma_{s}(0)
\end{equation}
where as usual $\sigma_{s}(u)$ is the sum of the $s$-th power of the divisors of $u$ if $u\neq 0$ and $\sigma_{s}(0)$ is normalized to be $\zeta(-s)/2$. His main theorem is to relate $\sum_{r,s}(n)$ to a close formula in terms of the $\Gamma$ and $\zeta$ functions as well as $\sigma_{r+s\pm 1}$. If $r+s=2,4,6,8$ or $12$, this close formula exactly calculates $\sum_{r,s}(n)$ but otherwise there is in addition an error term he denotes by $E_{r,s}(n)$.
He then goes on to observe that if $r+s=10,14,16,18,20$ or $24$, then $E_{r,s}(n)$ involves, as a function of $n$, only the constant $E_{r,s}(1)$, possibly $\sigma_{r+s-11}(n)$ and the function $\tau(n)$ defined by the $q$-expansion
\begin{equation}
\sum_{n=1}^{\infty}\tau(n)=q\prod_{n=1}^{\infty}(1-q^n)^{24}.
\end{equation}
When $r+s=10$, the simple formula $E_{r,s}(n)=E_{r,s}(1)\tau(n)$ obtains. Hence $\tau(n)$ appears in this original paper as a measure of the size of an error term. In particular, Ramanujan is extremely interested in bounding its size. He shows that $\tau(n)$ (and thus the error term) is not a $o(n^5)$, that it is a $O(n^7)$ and then conjectures (famously) that $\sum_{n=1}^{\infty}\tau(n)n^{-s}$ admits an Eulerian product and that $|\tau(n)|\leq n^{11/2}d(n)$ for all $n$. He goes on to note that this entails that $\tau(n)$ is not a $o(n^{11/2})$.
So there you have it: Ramanujan was interested in $\tau(n)$, in its arithmetical properties and in its size because it occurs as an error term in a formula he established to compute "convolutions" of the $\sigma$ functions. The two (or depending on the presentation three, but the original paper has two) statements called the Ramanujan conjecture directly bears on this question of estimating the error term; his central concern. From a modern perspective, we would probably follow the converse path, that is to say, we would note that logarithmically differentiating the $\Delta$ function yields many identities relating $\tau(n)$ and $\sigma(n)$, though that of course also originates from his computations. The paper is worth reading, but following Ramanujan's computations can prove a rough ride.