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 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 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. Dedekind^[2] 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.

Higher orders

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,

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

References

1. Leonard Eugene Dickson "History of the Theory Of Numbers", Vol. 1, p. 123, Chelsea Publishing 1952.
2. 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.

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