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:

:$f_p(x)=\sum_^\infty f(p^n)x^n.$

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:
:$f_p(x)=\frac.$

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 $\varphi_p(x)=\frac.$
* The multiplicative identity of the Dirichlet convolution $\delta$ has $\delta_p(x)=1.$
* The Liouville function $\lambda$ has $\lambda_p(x)=\frac.$
* The power function Id_''k'' has $(\textrm_k)_p(x)=\frac.$ Here, Id_''k'' is the completely multiplicative function $\operatorname_k(n)=n^k$.
* The divisor function $\sigma_k$ has $(\sigma_k)_p(x)=\frac.$
* The constant function, with value 1, satisfies $1_p(x) = (1-x)^$, i.e., is the geometric series.
* If $f(n) = 2^ = \sum_ \mu^2(d)$ is the power of the prime omega function, then $f_p(x) = \frac.$
* Suppose that ''f'' is multiplicative and ''g'' is any arithmetic function satisfying $f(p^) = f(p) f(p^n) - g(p) f(p^)$ for all primes ''p'' and $n \geq 1$. Then $f_p(x) = \left(1-f(p)x + g(p)x^2\right)^.$
* If $\mu_k(n) = \sum_ \mu_\left(\frac\right) \mu_\left(\frac\right)$ denotes the Möbius function of order ''k'', then $(\mu_k)_p(x) = \frac.$

See also

* Bell numbers

References

* {{Apostol IANT

Arithmetic functions
Series (mathematics)