Bell series

The following is a table of the Bell series of well-known arithmetic functions.

• The Möbius function ${\displaystyle \mu }$ has ${\displaystyle \mu _{p}(x)=1-x.}$
• The Mobius function squared has ${\displaystyle \mu _{p}^{2}(x)=1+x.}$
• Euler's totient ${\displaystyle \varphi }$ has ${\displaystyle \varphi _{p}(x)={\frac {1-x}{1-px}}.}$
• The multiplicative identity of the Dirichlet convolution ${\displaystyle \delta }$ has ${\displaystyle \delta _{p}(x)=1.}$
• The Liouville function ${\displaystyle \lambda }$ has ${\displaystyle \lambda _{p}(x)={\frac {1}{1+x}}.}$
• The power function Id_k has ${\displaystyle ({\textrm {Id}}_{k})_{p}(x)={\frac {1}{1-p^{k}x}}.}$ Here, Id_k is the completely multiplicative function ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}$.
• The divisor function ${\displaystyle \sigma _{k}}$ has ${\displaystyle (\sigma _{k})_{p}(x)={\frac {1}{(1-p^{k}x)(1-x)}}.}$
• The constant function, with value 1, satisfies ${\displaystyle 1_{p}(x)=(1-x)^{-1}}$, i.e., is the geometric series.
• If ${\displaystyle f(n)=2^{\omega (n)}=\sum _{d|n}\mu ^{2}(d)}$ is the power of the prime omega function, then ${\displaystyle f_{p}(x)={\frac {1+x}{1-x}}.}$
• Suppose that f is multiplicative and g is any arithmetic function satisfying ${\displaystyle f(p^{n+1})=f(p)f(p^{n})-g(p)f(p^{n-1})}$ for all primes p and ${\displaystyle n\geq 1}$. Then ${\displaystyle f_{p}(x)=\left(1-f(p)x+g(p)x^{2}\right)^{-1}.}$
• If ${\displaystyle \mu _{k}(n)=\sum _{d^{k}|n}\mu _{k-1}\left({\frac {n}{d^{k}}}\right)\mu _{k-1}\left({\frac {n}{d}}\right)}$ denotes the Möbius function of order k, then ${\displaystyle (\mu _{k})_{p}(x)={\frac {1-2x^{k}+x^{k+1}}{1-x}}.}$

• Bell numbers

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001