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

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

is:

1. 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... .

is a square-free number is the sum-of-divisors function.

, 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 the function as a Dirichlet convolution.

There is an additive definition of the psi function as well. Quoting from Dickson,

R. Dedekind then

is the totient function.

Higher orders

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

with Dirichlet series

1. .

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

1. .

If

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

1. .

References

External links

See also

• (page 25, equation (1))

• Section 3.13.2

• is 2, is 3, and is 4