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, ψ(1) is the empty product and so has value 1). The function was introduced by Richard Dedekind in connection with modular functions.

The value of ψ(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).

ψ(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 ψ(n) = σ(n).

The ψ function can also be defined by setting ψ(pⁿ) = (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 $\psi ={\mathrm {Id}}*|\mu |$ .

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)

\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

Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25, equation (1))
Carella, N. A. (2010). "Squarefree Integers And Extreme Values Of Some Arithmetic Functions". arXiv:1012.4817.
Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038. Section 3.13.2
A065958 is ψ₂, A065959 is ψ₃, and A065960 is ψ₄

External links

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

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