Bell series

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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

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

[edit] Examples

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

The Moebius function $μ$ has $μ p (x) = 1 - x .$
Euler's Totient $φ$ has $\phi_p(x)=\frac{1-x}{1-px}.$
The multiplicative identity of the Dirichlet convolution $δ$ has $δ p (x) = 1.$
The Liouville function $λ$ has $\lambda_p(x)=\frac{1}{1+x}.$
The power function Id_k has $(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.$ Here, Id_k is the completely multiplicative function $\operatorname{Id}_k(n)=n^k$ .
The divisor function $σ k$ has $(\sigma_k)_p(x)=\frac{1}{1-\sigma_k(p)x+p^kx^2}.$