Dedekind psi-function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

The Dedekind psi-function is an arithmetic function that sends any natural number $n$ to the number:

$\psi (n):=n\prod (1+\frac{1}{p_i})$

where the product is over all primes $p_i$ dividing $n$.

Relation with other arithmetic functions

• Euler phi-function has a similar expression to the Dedekind psi-function, except that the expression is $1-(1/p_i)$ rather than $1+(1/p_i)$.