# Bell series

In mathematics, the **Bell series** is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.

Given an arithmetic function and a prime , define the formal power series , called the Bell series of modulo as:

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:

Two series may be multiplied (sometimes called the *multiplication theorem*): For any two arithmetic functions and , let be their Dirichlet convolution. Then for every prime , one has:

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.

If is completely multiplicative, then formally:

## Examples

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 Id
_{k}has Here, Id_{k}is the completely multiplicative function . - The divisor function has