Is the inverse of a completely multiplicative function $f(n)$ with respect to Dirichlet convolution again completely multiplicative? I know that for multiplicative functions its true(Apostol's Analytic number theory page 36) but I think it doesn't hold for completely multiplicative functions. So i was looking for a counterexample(if it isn't true)...
2026-02-22 23:26:27.1771802787
Inverse of completely multiplicative function with respect to dirichlet convolution
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The Dirichlet inverse of the completely multiplicative constant $1$ function, i.e. $f(n) = 1$ for all $n$ is given by the Möbius function $\mu$ which is only multiplicative but not completely multiplicative.