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 and a prime
, define the formal power series
, called the Bell series of
modulo
as:
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:
In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.
If is completely multiplicative, then formally:
Examples
The following is a table of the Bell series of well-known arithmetic functions.
- The Möbius function
has
- Euler's Totient
has
- The multiplicative identity of the Dirichlet convolution
has
- The Liouville function
has
- The power function Idk has
Here, Idk is the completely multiplicative function
.
- The divisor function
has
See also
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, MR 0434929, Zbl 0335.10001