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 formally:

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

Examples

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

• The Möbius function $\mu$ has $\mu_p(x)=1-x.$
• Euler's Totient $\varphi$ has $\varphi_p(x)=\frac{1-x}{1-px}.$
• The multiplicative identity of the Dirichlet convolution $\delta$ has $\delta_p(x)=1.$
• The Liouville function $\lambda$ 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 $\sigma_k$ has $(\sigma_k)_p(x)=\frac{1}{1-(1+p^k) x+p^kx^2}.$

See also

• Bell numbers

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, MR 0434929, Zbl 0335.10001