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
and a prime
, define the formal power series
, called the Bell series of
modulo
as:
![{\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c42e940c92c984f7cbf75557d855d6442a332d48)
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
and
, one has
if and only if:
for all primes
.
Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions
and
, let
be their Dirichlet convolution. Then for every prime
, one has:
![{\displaystyle h_{p}(x)=f_{p}(x)g_{p}(x).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea5701452911f96d68edc226e07c2b8d6855b5d)
In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.
If
is completely multiplicative, then formally:
![{\displaystyle f_{p}(x)={\frac {1}{1-f(p)x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/388ae48f4d1e51a50358ec203ee3f55b548cc13d)
Examples[edit]
The following is a table of the Bell series of well-known arithmetic functions.
- The Möbius function
has ![{\displaystyle \mu _{p}(x)=1-x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4faf368337db77adbf9528eb07f60ea0a63785b6)
- The Mobius function squared has
![{\displaystyle \mu _{p}^{2}(x)=1+x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0201adb2dcf15b7d11e028a4fe39262573f31e26)
- Euler's totient
has ![{\displaystyle \varphi _{p}(x)={\frac {1-x}{1-px}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e14f8db1d5b5e8251cb64b882040f941cc0738b)
- The multiplicative identity of the Dirichlet convolution
has ![{\displaystyle \delta _{p}(x)=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4ed84baffc1730533e7c3878b20805a29adc0f)
- The Liouville function
has ![{\displaystyle \lambda _{p}(x)={\frac {1}{1+x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/323cce4d7a678fce15c4172c5e3fafd630a7f886)
- The power function Idk has
Here, Idk is the completely multiplicative function
. - The divisor function
has ![{\displaystyle (\sigma _{k})_{p}(x)={\frac {1}{(1-p^{k}x)(1-x)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d08303e4260e96d3fff4a9d14cbc72ed8872785b)
- The constant function, with value 1, satisfies
, i.e., is the geometric series. - If
is the power of the prime omega function, then ![{\displaystyle f_{p}(x)={\frac {1+x}{1-x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7be5592d260e5e42046b6dc20918290f04d675de)
- Suppose that f is multiplicative and g is any arithmetic function satisfying
for all primes p and
. Then ![{\displaystyle f_{p}(x)=\left(1-f(p)x+g(p)x^{2}\right)^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05644ec76cfa21a770fc270382fa5e573d4dd743)
- If
denotes the Möbius function of order k, then ![{\displaystyle (\mu _{k})_{p}(x)={\frac {1-2x^{k}+x^{k+1}}{1-x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5040fba9e403c1aa215ea400bb8e15fa74c695)
See also[edit]
References[edit]