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 $f$ 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

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

The Möbius function $\mu$ has $\mu _{p}(x)=1-x.$
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 $(\sigma _{k})_{p}(x)={\frac {1}{1-(1+p^{k})x+p^{k}x^{2}}}.$

References

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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Bell series

Examples

See also

References