Bell series

From Wikipedia, the free encyclopedia

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 <math>f</math> and a prime <math>p</math>, define the formal power series <math>f_p(x)</math>, called the Bell series of <math>f</math> modulo <math>p</math> as

<math>f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.</math>

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 <math>f</math> and <math>g</math>, one has <math>f=g</math> if and only if

<math>f_p(x)=g_p(x)</math> for all primes <math>p</math>.

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions <math>f</math> and <math>g</math>, let <math>h=f*g</math> be their Dirichlet convolution. Then for every prime <math>p</math>, one has

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.

If <math>f</math> is completely multiplicative, then

[edit] Examples

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

The Moebius function <math>\mu</math> has <math>\mu_p(x)=1-x.</math>
Euler's Totient <math>\phi</math> has <math>\phi_p(x)=\frac{1-x}{1-px}.</math>
The identity function <math>I</math> has <math>I_p(x)=1.</math>
The Liouville function <math>\lambda</math> has <math>\lambda_p(x)=\frac{1}{1+x}.</math>
The power function Id_k has <math>(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.</math> Here, Id_k is the completely multiplicative function <math>\operatorname{Id}_k(n)=n^k</math>.
The divisor function <math>\sigma_k</math> has <math>(\sigma_k)_p(x)=\frac{1}{1-\sigma_k(p)x+p^kx^2}.</math>