Dedekind psi function

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

ψ ( n ) = n ∏ p | n ( 1 + 1 p ) ,

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 the 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)p^n-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

∑ ψ ( n ) n s = ζ ( s ) ζ ( s − 1 ) ζ ( 2 s ) .

This is also a consequence of the fact that we can write as a Dirichlet convolution of ψ = I d ∗ | μ | .