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 $$\mu_p(x)=1-x.$$
 * 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 Idk has $$(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.$$ Here, Idk is the completely multiplicative function  $$\operatorname{Id}_k(n)=n^k$$.
 * The divisor function $$\sigma_k$$ has $$(\sigma_k)_p(x)=\frac{1}{(1-p^kx)(1-x)}.$$
 * 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 $$(\mu_k)_p(x) = \frac{1-2x^k+x^{k+1}}{1-x}.$$