The Ramanujan tau function $\tau$ is given by the coefficients of the power series expansion of the modular discriminant $\Delta$: $$ \sum_{n=1}^{+\infty} \tau(n)\,q^n := q\,\prod_{i=1}^{+\infty}(1-q^i)^{24} =: \Delta(z) $$ (where $q = \exp(2i\pi z)$). This function has lots of remarkable properties, inter alia:
it is multiplicative, viꝫ. $\tau(mn) = \tau(m)\,\tau(n)$ when $\operatorname{gcd}(m,n)=1$,
$\tau(p^{r+1}) = \tau(p)\,\tau(p^r) - p^{11}\,\tau(p^{r-1})$ for $p$ prime and $r\geq 1$,
$|\tau(p)| \leq 2 p^{11/2}$ for all primes $p$,
which are connected to the modular form properties of $\Delta$.
However, to get a sense of how remarkable these properties are, I would like to know what happens for “other values of $24$” (it's irritating that nobody gives this even a passing mention). In other words, for $k\geq 1$, let us define: $$ \sum_{n=1}^{+\infty} \tau_k(n)\,q^n := q\,\prod_{i=1}^{+\infty}(1-q^i)^k $$
Questions:
Is $24$ the only value of $k$ for which $\tau_k$ is multiplicative?
How does $\tau_k(n)$ grow asymptotically for fixed $k$ and large $n$? (I'll accept a heuristic here!)
Generally speaking, what properties does $\tau_k$ possess for any $k$ and in what sense is $\tau_{24}$ completely unusual among the $\tau_k$? I'd like to get a big picture, here.
