One motivation I have seen given for Ramanujan's conjecture for the estimate $$ |a_p|< C p^{k - \frac{1}{2}} $$ for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to show that the error term in some formulas for the number of representation of a number by a certain quadratic form (which comes from a cusp form) is dominated by the main term. However, at least in applications in Serre's "a course in arithmetic", it seems that even the Hecke estimate suffices. I know that this is not the case for some more recent applications (say, construction of Ramanujan graphs, `a la Lubotzky-Philip-Sarnak), but these could have hardly been Ramanujan's motivation.

I would like to know what was Ramanujan's original motivation for making this conjecture, and also a hint as to how he may have come up with the $1/2$ power saving term. This question came up in a course on analytic number theory I taught this semester. ANT is not my field of research, so I'd not be surprised if the answer to this question turns out to be well-known or even trivial. I will still appreciate it if an expert could respond or provide a reference.

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    $\begingroup$ Have you looked at Ramanujan's original paper "On certain arithmetical functions"? He writes down ("For it appears that") the Hecke relations (as a relation connecting the Dirichlet series for $\tau(n)$ with an Euler product). Then he says what this means for $\tau(p^k)$ if one writes $\tau(p)$ as $2p^{11/2} \cos \theta_p$, and this substitution together with his calculations immediately would suggest that $\theta_p$ is always real, which was his conjecture. $\endgroup$ – Lucia May 6 '16 at 17:48

Ramanujan made his conjecture before Hecke gave his bound. As Lucia noted, the conjecture was based on empirical computation concerning the coefficients $\tau(n)$ of the $\Delta$ modular form, so Ramanujan would have had no reason to propose a conjecture less precise than what was apparently true.

In other words, no one came up with the 1/2 power saving term. Ramanujan found directly the best possible bound, and the first one who proved something approaching, Hecke, was off by 1/2.

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    $\begingroup$ Maybe it is "no one came up with.." $\endgroup$ – kodlu May 6 '16 at 23:42
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    $\begingroup$ Joël, you're right of course. But Ramanujan himself did know of somewhat weaker bounds for $\tau(n)$: namely $\tau(n)=O(n^7)$. His paper starts out by multiplying Eisenstein series of weights $r+1$ and $s+1$ and comparing that with Eisenstein series of weight $r+s+2$, and he thinks of $\tau(n)$ at least partly as the error term in this approximation (which he notes as an exact formula for small $r$ and $s$). Subsequently he also ties this all up with representations of numbers by sums of squares. $\endgroup$ – Lucia May 7 '16 at 3:44

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