I am afraid to continue to ask trivial things but really I do not know how to proceed, so I ask the experts:

A specially multiplicative function, is a function $f$ from positive integers to complex numbers, that satisfies: $$ f(n)f(m) = \sum_{d \mid \gcd(m,n)} f(\frac{mn}{d^2})g(d) $$ for all $m,n$, where $g$ is a completely multiplicative function, i;e. $$ g(rs) = g(r)g(s) $$ for all positive integers $r,s.$

Seems well known that any specially multiplicative function $F$ is the Dirichlet convolution of $2$ completely multiplicative functions. (We write, say, $F = r * s$). That is:

$$ F(n) = r * s (n) = \sum_{d \mid n} r(d) s(n/d) $$ with $r,s$ completely multiplicative functions.

I checked that

$$ \sigma = id * z $$

where $\sigma$ is the sum of divisors's function, (that is specially multiplicative with a explicit $g$), $$ id(n) = n $$ for all $n$ and $$ z(n) = 1 $$ for all $n.$

But I am (unfortunately) unable to find completely multiplicative functions $a,b$ such that

$$ \tau = a * b $$

where $\tau$ is Ramanujan's tau function.

Question 1 : Somebody can display such $a,b$ ???

Probably, there is a standard construction of such $a,b$ for any specially multiplicative function.

Question 2 : What is the recipe to get such $a,b$ for a given specially multiplicative function ???