# 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:

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

## Examples

The following is a table of the Bell series of well-known arithmetic functions.

• The Möbius function ${\displaystyle \mu }$ has ${\displaystyle \mu _{p}(x)=1-x.}$
• Euler's Totient ${\displaystyle \varphi }$ has ${\displaystyle \varphi _{p}(x)={\frac {1-x}{1-px}}.}$
• The multiplicative identity of the Dirichlet convolution ${\displaystyle \delta }$ has ${\displaystyle \delta _{p}(x)=1.}$
• The Liouville function ${\displaystyle \lambda }$ has ${\displaystyle \lambda _{p}(x)={\frac {1}{1+x}}.}$
• The power function Idk has ${\displaystyle ({\textrm {Id}}_{k})_{p}(x)={\frac {1}{1-p^{k}x}}.}$ Here, Idk is the completely multiplicative function ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}$.
• The divisor function ${\displaystyle \sigma _{k}}$ has ${\displaystyle (\sigma _{k})_{p}(x)={\frac {1}{1-(1+p^{k})x+p^{k}x^{2}}}.}$