bell series

In mathematics, the Bell series is a formal power series used to study properties of multiplicative 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

f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.

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

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

f_p(x)=\frac{1}{1-f(p)x}.

Examples

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

The Moebius function $\mu$ has $\mu_p(x)=1-x.$
Euler's Totient $\phi$ has $\phi_p(x)=\frac{1-x}{1-px}.$
The identity function $I$ has $I_p(x)=1.$
The Liouville function $\lambda$ has $\lambda_p(x)=\frac{1}{1+x}.$
The power function Id_k has $(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.$
The divisor function $\sigma_k$ has $(\sigma_k)_p(x)=\frac{1}{1-\sigma_k(p)x+p^kx^2}.$

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York

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