Bell series information

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}}.

Bell series information

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