Bell series - Definition and Overview

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>

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>.

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

<math>h_p(x)=f_p(x) g_p(x).\,<math>

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

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

<math>f_p(x)=\frac{1}{1-f(p)x}.<math>

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>
• The divisor function <math>\sigma_k<math> has <math>(\sigma_k)_p(x)=\frac{1}{1-\sigma_k(p)x+p^kx^2}<math>

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York