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:

$$(f*g)(n) \ =\ \sum_{d\,\mid \,n} f(d)\,g\!\left(\frac{n}{d}\right) \ =\ \sum_{ab\,=\,n}\!f(a)\,g(b)$$

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

Important functions

• $\varepsilon$ - Unit function

• $\mu$ - Mobius function

• $1 * \mu = \varepsilon$