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:
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