Dedekind psi function

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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 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)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

\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[edit]

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\ge 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[edit]

  • 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
  • OEISA065958 is ψ2, OEISA065959 is ψ3, and OEISA065960 is ψ4

External links[edit]