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 f and a prime p, define the formal power series f_p(x), called the Bell series of f modulo p 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 f and g, one has f=g if and only if: Two series may be multiplied (sometimes called the 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: In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. If f is completely multiplicative, then formally:

Examples

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