In mathematics, the Bell series is a formal power series used to study properties of multiplicative 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

Uniqueness theorem. Given multiplicative functions f and g, one has f = g if and only if

f_p(x) = g_p(x) for all primes p.

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

Examples

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

• The Moebius function μ has μ_p(x) = 1 - x.
• Euler's Totient φ has
• The identity function I has I_p(x) = 1.
• The Liouville function λ has
• The power function Id_k has
• The divisor function σ_k has

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York