Bell series

In mathematics, the Bell series is a formal power series used to study properties of multiplicative 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 series 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

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

Examples

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

• The Moebius function $\mu$ has $\mu _{p}(x)=1-x.$
• Euler's Totient $\phi$ has $\phi _{p}(x)={\frac {1-x}{1-px}}.$
• The identity function $I$ has $I_{p}(x)=1.$
• 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 $(\sigma _{k})_{p}(x)={\frac {1}{1-\sigma _{k}(p)x+p^{k}x^{2}}}.$

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639