Dedekind Psi Function

The Dedekind Psi function is closely related to the Euler phi function by the formula

ψ (n) = \frac{J_{2} (n)}{J_{1} (n)} = \frac{J_{2} (n)}{ϕ (n)}

and we can write it as

ψ = Id * | μ |

and it can be written as a product

ψ (n) = n \prod_{p ∣ n} (1 + \frac{1}{p})

There are higher order via ratios of the Jordan totient function

ψ_{k} (n) = \frac{J_{2 k} (n)}{J_{k} (n)}

and can be written as a convolution

ψ_{k} = {Id}_{k} * | μ |