Dedekind psi function

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

${\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right),}$

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

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

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the OEIS).

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

The ${\displaystyle \psi }$ function can also be defined by setting ${\displaystyle \psi (p^{n})=(p+1)p^{n-1}}$ for powers of any prime ${\displaystyle p}$, 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

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

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

There is an additive definition of the psi function as well. Quoting from Dickson,¹

R. Dedekind² proved that, if ${\displaystyle n}$ is decomposed in every way into a product ${\displaystyle ab}$ and if ${\displaystyle e}$ is the g.c.d. of ${\displaystyle a,b}$ then

${\displaystyle \sum _{a}(a/e)\varphi (e)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right)}$

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

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

${\displaystyle \psi _{k}(n)={\frac {J_{2k}(n)}{J_{k}(n)}}}$

with Dirichlet series

${\displaystyle \sum _{n\geq 1}{\frac {\psi _{k}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-k)}{\zeta (2s)}}}$.

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

${\displaystyle \psi _{k}(n)=n^{k}*\mu ^{2}(n)}$.

If

${\displaystyle \epsilon _{2}=1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots }$

is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function,

${\displaystyle \epsilon _{2}(n)*\psi _{k}(n)=\sigma _{k}(n)}$.