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: The following is a table of the Bell series of well-known arithmetic functions. The Möbius function has The Mobius function squared 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 The constant function, with value 1, satisfies , i.e., is the geometric series. If is the power of the prime omega function, then Suppose that f is multiplicative and g is any arithmetic function satisfying for all primes p and .