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:

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

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 ${\mu }_p(x)=1-x.$
• The Mobius function squared has ${\mu }_p^2(x)=1+x.$
• Euler's totient $\phi$ has ${\phi }_p(x)=\frac{1-x}{1-px}.$
• 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 $({\text{<mrow data-mjx-texclass="ORD">Id</mrow>}}_k{)}_p(x)=\frac{1}{1-p^{k}x}.$ Here, Id_k is the completely multiplicative function ${\mathrm{Id}}_k(n)=n^k$.
• The divisor function ${\sigma }_k$ has $({\sigma }_k{)}_p(x)=\frac{1}{(1-p^{k}x)(1-x)}.$
• The constant function, with value 1, satisfies $1_p(x)=(1-x{)}^{-1}$, i.e., is the geometric series.
• If $f(n)=2^{\omega (n)}=\sum _{d|n}{\mu }^2(d)$ is the power of the prime omega function, then $f_p(x)=\frac{1+x}{1-x}.$
• Suppose that f is multiplicative and g is any arithmetic function satisfying $f(p^{n+1})=f(p)f(p^n)-g(p)f(p^{n-1})$ for all primes p and $n\ge 1$. Then $f_p(x)={(1-f(p)x+g(p)x^2)}^{-1}.$
• If ${\mu }_k(n)=\sum _{d^k|n}{\mu }_{k-1}\left(\frac{n}{d^k}\right){\mu }_{k-1}\left(\frac{n}{d}\right)$ denotes the Möbius function of order k, then $({\mu }_k{)}_p(x)=\frac{1-2x^k+x^{k+1}}{1-x}.$

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