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 ${\displaystyle f}$ and a prime ${\displaystyle p}$, define the formal power series ${\displaystyle f_{p}(x)}$, called the Bell series of ${\displaystyle f}$ modulo ${\displaystyle p}$ as:

${\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}.}$

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 ${\displaystyle f}$ and ${\displaystyle g}$, one has ${\displaystyle f=g}$ if and only if:

${\displaystyle f_{p}(x)=g_{p}(x)}$ for all primes ${\displaystyle p}$.

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions ${\displaystyle f}$ and ${\displaystyle g}$, let ${\displaystyle h=f*g}$ be their Dirichlet convolution. Then for every prime ${\displaystyle p}$, one has:

${\displaystyle h_{p}(x)=f_{p}(x)g_{p}(x).\,}$

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

If ${\displaystyle f}$ is completely multiplicative, then formally:

${\displaystyle f_{p}(x)={\frac {1}{1-f(p)x}}.}$