Dedekind psi function

In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by


 * $$ \psi(n) = n \prod_{p|n}\left(1+\frac{1}{p}\right),$$

where the product is taken over all primes $$p$$ dividing $$n.$$ (By convention, $$\psi(1)$$, which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions.

The value of $$\psi(n)$$ for the first few integers $$n$$ is:


 * 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ....

The function $$\psi(n)$$ is greater than $$n$$ for all $$n$$ greater than 1, and is even for all $$n$$ greater than 2. If $$n$$ is a square-free number then $$\psi(n) = \sigma(n)$$, where $$\sigma(n)$$ is the divisor function.

The $$\psi$$ function can also be defined by setting $$\psi(p^n) = (p+1)p^{n-1}$$ for powers of any prime $$p$$, and then extending the definition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is


 * $$\sum \frac{\psi(n)}{n^s} = \frac{\zeta(s) \zeta(s-1)}{\zeta(2s)}.$$

This is also a consequence of the fact that we can write as a Dirichlet convolution of $$\psi= \mathrm{Id} * |\mu| $$.

There is an additive definition of the psi function as well. Quoting from Dickson,

R. Dedekind proved that, if $$n$$ is decomposed in every way into a product $$ab$$ and if $$e$$ is the g.c.d. of $$a, b$$ then
 * $$\sum_{a} (a/e) \varphi(e) = n \prod_{p|n}\left(1+\frac{1}{p}\right)$$

where $$a$$ ranges over all divisors of $$n$$ and $$p$$ over the prime divisors of $$n$$ and $$\varphi$$ is the totient function.

Higher orders
The generalization to higher orders via ratios of Jordan's totient is


 * $$\psi_k(n)=\frac{J_{2k}(n)}{J_k(n)}$$

with Dirichlet series


 * $$\sum_{n\ge 1}\frac{\psi_k(n)}{n^s} = \frac{\zeta(s)\zeta(s-k)}{\zeta(2s)}$$.

It is also the Dirichlet convolution of a power and the square of the Möbius function,


 * $$\psi_k(n) = n^k * \mu^2(n)$$.

If


 * $$\epsilon_2 = 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots$$

is the characteristic function of the squares, another Dirichlet convolution leads to the generalized &sigma;-function,


 * $$\epsilon_2(n) * \psi_k(n) = \sigma_k(n)$$.