Dedekind psi function

In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by

where the product is taken over all primes dividing (by convention, is the empty product and so has value 1). The function was introduced by Richard Dedekind in connection with modular functions.

The value of for the first few integers is:

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24 ... (sequence A001615 in the OEIS).

The function is greater than for all greater than 1, and is even for all greater than 2. If is a square-free number then , where is the divisor function.

The function can also be defined by setting for powers of any prime , 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 .

There is an additive definition of the PSI function as well. Quoting from[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

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

note that is the totient function.

Higher ordersEdit

The generalization to higher orders via ratios of Jordan's totient is


with Dirichlet series


It is also the Dirichlet convolution of a power and the square of the Möbius function,




is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function,



  • 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 linksEdit

  • ^ Leonard Eugene Dickson "History Of The Theory Of Numbers", Vol 1, p123 Chelsea Publishing 1952
  • ^ Journal für die reine und angewandte Mathematik,vol 83, 1877, p288. CF H Weber, Elliptische Functionen ,1901,244-5; ed 2, 1008(Algebra III), 234-5