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 : 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.
