TITLE: Is inverse of Dirichlet convolution unique ?
QUESTION [7 upvotes]: We know that the Dirichlet inverse of the constant function 1 is the Möbius function.
I just want to confirm that Möbius function is the only function which is the inverse of constant function for Dirichlet convolution.
Can anyone confirm this ?
Thank you.

REPLY [5 votes]: Suppose that $g_1$ and $g_2$ are two inverses of $f$. Then we have:
$$g_1 = g_1*e = g_1*(f*g_2) = (g_1*f)*g_2 = e*g_2 = g_2 $$
Using only the properties of the convolution product.