Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem. Given multiplicative functions and , one has if and only if:
- for all primes .
In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.
If is completely multiplicative, then:
The following is a table of the Bell series of well-known arithmetic functions.
- The Möbius function has
- Euler's Totient has
- The multiplicative identity of the Dirichlet convolution has
- The Liouville function has
- The power function Idk has Here, Idk is the completely multiplicative function .
- The divisor function has