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 [math]\displaystyle{ f }[/math] and a prime [math]\displaystyle{ p }[/math], define the formal power series [math]\displaystyle{ f_p(x) }[/math], called the Bell series of [math]\displaystyle{ f }[/math] modulo [math]\displaystyle{ p }[/math] as:
- [math]\displaystyle{ f_p(x)=\sum_{n=0}^\infty f(p^n)x^n. }[/math]
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 [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math], one has [math]\displaystyle{ f=g }[/math] if and only if:
- [math]\displaystyle{ f_p(x)=g_p(x) }[/math] for all primes [math]\displaystyle{ p }[/math].
Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math], let [math]\displaystyle{ h=f*g }[/math] be their Dirichlet convolution. Then for every prime [math]\displaystyle{ p }[/math], one has:
- [math]\displaystyle{ 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]\displaystyle{ f }[/math] is completely multiplicative, then formally:
- [math]\displaystyle{ 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 Möbius function [math]\displaystyle{ \mu }[/math] has [math]\displaystyle{ \mu_p(x)=1-x. }[/math]
- The Mobius function squared has [math]\displaystyle{ \mu_p^2(x) = 1+x. }[/math]
- Euler's totient [math]\displaystyle{ \varphi }[/math] has [math]\displaystyle{ \varphi_p(x)=\frac{1-x}{1-px}. }[/math]
- The multiplicative identity of the Dirichlet convolution [math]\displaystyle{ \delta }[/math] has [math]\displaystyle{ \delta_p(x)=1. }[/math]
- The Liouville function [math]\displaystyle{ \lambda }[/math] has [math]\displaystyle{ \lambda_p(x)=\frac{1}{1+x}. }[/math]
- The power function Idk has [math]\displaystyle{ (\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}. }[/math] Here, Idk is the completely multiplicative function [math]\displaystyle{ \operatorname{Id}_k(n)=n^k }[/math].
- The divisor function [math]\displaystyle{ \sigma_k }[/math] has [math]\displaystyle{ (\sigma_k)_p(x)=\frac{1}{(1-p^kx)(1-x)}. }[/math]
- The constant function, with value 1, satisfies [math]\displaystyle{ 1_p(x) = (1-x)^{-1} }[/math], i.e., is the geometric series.
- If [math]\displaystyle{ f(n) = 2^{\omega(n)} = \sum_{d|n} \mu^2(d) }[/math] is the power of the prime omega function, then [math]\displaystyle{ f_p(x) = \frac{1+x}{1-x}. }[/math]
- Suppose that f is multiplicative and g is any arithmetic function satisfying [math]\displaystyle{ f(p^{n+1}) = f(p) f(p^n) - g(p) f(p^{n-1}) }[/math] for all primes p and [math]\displaystyle{ n \geq 1 }[/math]. Then [math]\displaystyle{ f_p(x) = \left(1-f(p)x + g(p)x^2\right)^{-1}. }[/math]
- If [math]\displaystyle{ \mu_k(n) = \sum_{d^k|n} \mu_{k-1}\left(\frac{n}{d^k}\right) \mu_{k-1}\left(\frac{n}{d}\right) }[/math] denotes the Möbius function of order k, then [math]\displaystyle{ (\mu_k)_p(x) = \frac{1-2x^k+x^{k+1}}{1-x}. }[/math]
See also
- Bell numbers
References
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3
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