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 f and a prime p, define the formal power series f_p(x), called the Bell series of f modulo p as:

$f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.$

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:

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

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:

$h_p(x)=f_p(x) g_p(x).\,$

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.

If f is completely multiplicative, then:

$f_p(x)=\frac{1}{1-f(p)x}.$

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 $\varphi_p(x)=\frac{1-x}{1-px}.$
• The multiplicative identity of the Dirichlet convolution δ has δ_p(x) = 1.
• The Liouville function λ has $\lambda_p(x)=\frac{1}{1+x}.$
• The power function Id_k has $(\textrm{Id}_k)_p(x)=\frac{1}{1-p^kx}.$ Here, Id_k is the completely multiplicative function $\operatorname{Id}_k(n)=n^k$.
• The divisor function σ_k has $(\sigma_k)_p(x)=\frac{1}{1-(1+p^k) x+p^kx^2}.$

References

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929