Английская Википедия: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>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 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>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:
- <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 formally:
- <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 Möbius function <math>\mu</math> has <math>\mu_p(x)=1-x.</math>
- The Mobius function squared has <math>\mu_p^2(x) = 1+x.</math>
- Euler's totient <math>\varphi</math> has <math>\varphi_p(x)=\frac{1-x}{1-px}.</math>
- The multiplicative identity of the Dirichlet convolution <math>\delta</math> has <math>\delta_p(x)=1.</math>
- The Liouville function <math>\lambda</math> has <math>\lambda_p(x)=\frac{1}{1+x}.</math>
- The power function Idk has <math>(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.</math> Here, Idk 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-p^kx)(1-x)}.</math>
- The constant function, with value 1, satisfies <math>1_p(x) = (1-x)^{-1}</math>, i.e., is the geometric series.
- If <math>f(n) = 2^{\omega(n)} = \sum_{d|n} \mu^2(d)</math> is the power of the prime omega function, then <math>f_p(x) = \frac{1+x}{1-x}.</math>
- Suppose that f is multiplicative and g is any arithmetic function satisfying <math>f(p^{n+1}) = f(p) f(p^n) - g(p) f(p^{n-1})</math> for all primes p and <math>n \geq 1</math>. Then <math>f_p(x) = \left(1-f(p)x + g(p)x^2\right)^{-1}.</math>
- If <math>\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>(\mu_k)_p(x) = \frac{1-2x^k+x^{k+1}}{1-x}.</math>
See also
References
