Dirichlet Convolution

If $f,g:\mathbb{N}\to \mathbb{C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined by:

$$(f\ast g)(n)=\sum _{d\mid n}f(d)g\left(\frac{n}{d}\right)=\sum _{ab=n}f(a)g(b)$$

• Commutative: $f\ast g=g\ast f$
• Associative: $(f\ast g)\ast h=f\ast (g\ast h)$
• Distributive over pointwise addition: $f\ast (g+h)=f\ast g+f\ast h$
  • Pointwise addition: $f+g$ is defined by $(f+g)(n)=f(n)+g(n)$
• Multiplicative Identity: $f\ast \epsilon =\epsilon \ast f=f$
  • $\epsilon$ is the unit function defined as
    $$\epsilon (n)=\{\begin{array}{ll}1,& \text{if }n=1\\ 0,& \text{if }n\ne 1\end{array}$$
• Multiplicative Inverse: for each $f$ having $f(1)\ne 0$, there exists an arithmetic function $f^{-1}$ with $f\ast f^{-1}=\epsilon$, called the Dirichlet inverse of $f$

Important functions

• $\epsilon$ - Unit function

• $\mu$ - Mobius function

• $1\ast \mu =\epsilon$