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:

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

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

• The Möbius function ${\displaystyle \mu }$ has ${\displaystyle \mu _{p}(x)=1-x.}$
• The Mobius function squared has ${\displaystyle \mu _{p}^{2}(x)=1+x.}$
• 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 Id_k has ${\displaystyle ({\textrm {Id}}_{k})_{p}(x)={\frac {1}{1-p^{k}x}}.}$ Here, Id_k is the completely multiplicative function ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}$.
• The divisor function ${\displaystyle \sigma _{k}}$ has ${\displaystyle (\sigma _{k})_{p}(x)={\frac {1}{(1-p^{k}x)(1-x)}}.}$
• The constant function, with value 1, satisfies ${\displaystyle 1_{p}(x)=(1-x)^{-1}}$, i.e., is the geometric series.
• If ${\displaystyle f(n)=2^{\omega (n)}=\sum _{d|n}\mu ^{2}(d)}$ is the power of the prime omega function, then ${\displaystyle f_{p}(x)={\frac {1+x}{1-x}}.}$
• Suppose that f is multiplicative and g is any arithmetic function satisfying ${\displaystyle f(p^{n+1})=f(p)f(p^{n})-g(p)f(p^{n-1})}$ for all primes p and ${\displaystyle n\geq 1}$. Then ${\displaystyle f_{p}(x)=\left(1-f(p)x+g(p)x^{2}\right)^{-1}.}$
• If ${\displaystyle \mu _{k}(n)=\sum _{d^{k}|n}\mu _{k-1}\left({\frac {n}{d^{k}}}\right)\mu _{k-1}\left({\frac {n}{d}}\right)}$ denotes the Möbius function of order k, then ${\displaystyle (\mu _{k})_{p}(x)={\frac {1-2x^{k}+x^{k+1}}{1-x}}.}$

• Bell numbers

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001