I am interested in finding out a method to invert Dirichlet selfconvolution.
In math expressions it means: Find out $a$ once $b=a*a$ is known
So a kind of squareroot of the Dirichlet product.
I also found that Mobius inversion formula clearly states an inversion for $b=a*1$:
https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
but I found no similar result for selfconvolution.
Assume $a(1)\ne 0$.
By induction once you know $a(1)$ and $a(d)$ for $d| n, d<n$ and $\sum_{d| n} a(d)a(n/d)$ then you know $a(n)$. Thus the only indeterminate is $a(1)$ which is one of the two square roots of $b(1)$.
If $a(1)=0$ it works the same way except we are dealing with the generalized Dirichlet series $(\sum_{n\ge N} a(n)(n/N)^{-s})^2= \sum_{n\ge N^2} b(n)(n/N^2)^{-s}$