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.
Suppose that f is multiplicative and g is any arithmetic function satisfying for all primes p and . Then
If denotes the , then
See also[]
Bell numbers
References[]
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN978-0-387-90163-3, MR0434929, Zbl0335.10001