Dedekind psi function

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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, , which is the empty product, 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 Dickson,[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 orders[edit]

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,

.

If

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

.

References[edit]

  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[edit]

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

See also[edit]