Dedekind psi function
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:
The ψ function can also be defined by setting ψ(pn) = (p+1)pn-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
This is also a consequence of the fact that we can write as a Dirichlet convolution of .
The generalization to higher orders via ratios of Jordan's totient is
with Dirichlet series
- 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