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 positive integers $n$ is:

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

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 sum-of-divisors 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 the function 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

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

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

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

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 σ-function,

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

External links

• Weisstein, Eric W. "Dedekind Function". MathWorld.

See also

• Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25, equation (1))
• Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT]. Section 3.13.2
• OEIS: A065958 is ψ₂, OEIS: A065959 is ψ₃, and OEIS: A065960 is ψ₄