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, ... (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 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,^[1]

R. Dedekind^[2] 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

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

where a ranges over all divisors of n and p over the prime divisors of n.

Note that $\Phi$ 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)

.

References

^ Leonard Eugene Dickson "History of the Theory Of Numbers", Vol. 1, p. 123, Chelsea Publishing 1952.
^ Journal 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

External links

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

Dedekind psi function

Higher orders

References

External links

See also