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
, define the formal power series
, called the Bell series of
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
if and only if: Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions
, one has: In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.
is completely multiplicative, then formally: The following is a table of the Bell series of well-known arithmetic functions.