The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.

Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$?

**Context**

The partition function $p$, defined by
$$\prod_{n=1}^\infty \frac{1}{1-q^n}=\sum_{n=0}^\infty p(n)q^n$$
has a known asymptotic formula, namely
$$p (n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right);$$
a proof of this uses elliptic/modular function theory.
It was quite shocking for me to observe that no asymptotic formula for $\tau (n)$ appears in the OEIS, but it is maybe hiding somewhere. The changes of signs of $\tau$ appear quite chaotic; still, one would wish to find an asymptotic formula for $|\tau (n)|$. We know that
$$|\tau (n)|=O(n^{\frac{11}{2}+\epsilon})$$
and
$$|\tau (p)|\le 2p^{\frac{11}{2}}$$
if $p$ is prime.
According to Zagier,
>The proof of these formulas, if written out from scratch, has been estimated at 2000 pages.

and in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.