An arithmetic function $f: \mathbb{N}_0 \to \mathbb{C}$ is said to be completely multiplicative if $f(mn)=f(m)f(n)$, for all $m,n \geq 0$. The Dirichlet convolution of two arithmetic functions $f,g$ is defined as:
$$(f \ast g)(n)= \sum_{mm'=n}f(m)g(m').$$
Is the Dirichlet convolution of two completely multiplicative arithmetic functions necessarily completely multiplicative?
Best Answer
It is completely multiplicative iff $f(p)g(p)=0$ for all $p$. Completely multiplicative means $$\sum_n h(n)n^{-s}=\prod_p \frac1{1-h(p)p^{-s}}$$