Dedekind psi function

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

:$\psi(n) = n \prod_\left(1+\frac\right),$

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 $\psi(p^n) = (p+1)p^$ 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

:$\sum \frac = \frac.$

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

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

R. DedekindJournal für die reine und angewandte Mathematik, vol. 83, 1877, p. 288. Cf. H. Weber, Elliptische Functionen, 1901, 244-5; ed. 2, 1008 (Algebra III), 234-5 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

:$\sum_ (a/e) \varphi(e) = n \prod_\left(1+\frac\right)$

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

:$\psi_k(n)=\frac$

with Dirichlet series

:$\sum_\frac = \frac$.

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,

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

References

External links

*

See also

* (page 25, equation (1))
* Section 3.13.2
* is ψ₂, is ψ₃, and {{OEIS2C, A065960 is ψ₄

Multiplicative functions