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

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 $f$ and $g$ , one has $f=g$ if and only if:

f_{p}(x)=g_{p}(x)

for all primes

p

.

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

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 $f$ is completely multiplicative, then formally:

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

Examples[edit]

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

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

References[edit]

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

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