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 p {\displaystyle p} dividing n . {\displaystyle n.} (By convention, ψ ( 1 ) {\displaystyle \psi (1)} , which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions.

The value of ψ ( n ) {\displaystyle \psi (n)} for the first few integers n {\displaystyle n} is:

The function ψ ( n ) {\displaystyle \psi (n)} is greater than n {\displaystyle n} for all n {\displaystyle n} greater than 1, and is even for all n {\displaystyle n} greater than 2. If n {\displaystyle n} is a square-free number then ψ ( n ) = σ ( n ) {\displaystyle \psi (n)=\sigma (n)} , where σ ( n ) {\displaystyle \sigma (n)} is the sum-of-divisors function.

The ψ {\displaystyle \psi } function can also be defined by setting ψ ( p n ) = ( p + 1 ) p n − 1 {\displaystyle \psi (p^{n})=(p+1)p^{n-1}} for powers of any prime p {\displaystyle 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

This is also a consequence of the fact that we can write as a Dirichlet convolution of ψ = I d ∗ | μ | {\displaystyle \psi =\mathrm {Id} *|\mu |} .

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

R. Dedekind proved that, if n {\displaystyle n} is decomposed in every way into a product a b {\displaystyle ab} and if e {\displaystyle e} is the g.c.d. of a , b {\displaystyle a,b} then

where a {\displaystyle a} ranges over all divisors of n {\displaystyle n} and p {\displaystyle p} over the prime divisors of n {\displaystyle n} and φ {\displaystyle \varphi } is the totient function.