Dirichlet Convolution - Examples

Examples

In these formulas

is the multiplicative identity. (I.e. (1) = 1, all other values 0.)

1 is the constant function whose value is 1 for all n. (I.e. 1(n) = 1.) Keep in mind that 1 is not the identity.

1_C, where is a set is the indicator function. (I.e. 1_C(n) = 1 if n ∈ C, 0 otherwise.)

Id is the identity function whose value is n. (I.e. Id(n) = n.)

Id_k is the kth power function. (I.e. Id_k(n) = nk.)

The other functions are defined in the article arithmetical function.

• 1 * μ = (the Dirichlet inverse of the constant function 1 is the Möbius function.) This implies

• g = f * 1 if and only if f = g * μ (the Möbius inversion formula).

• λ * |μ| = where λ is Liouville's function.

• λ * 1 = 1_Sq where Sq = {1, 4, 9, ...} is the set of squares

• _k = Id_k * 1 definition of the function σ_k

• = Id * 1 definition of the function σ = σ₁

• d = 1 * 1 definition of the function d(n) = σ₀

• Id_k = _k * Möbius inversion of the formulas for σ_k, σ, and d.

• Id = *

• 1 = d * μ

• d 3 * 1 = (d * 1)2

• * 1 = Id This formula is proved in the article Euler's totient function.

• J_k * 1 = Id_k

• (Id_sJ_r) * J_s = J_{s + r}

• = * d Proof: convolve 1 to both sides of Id = * 1.

• Λ * 1 = log where Λ is von Mangoldts' function