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 fp(x), called the Bell series of f modulo p as:

fp(x)=∑n=0∞f(pn)xn.

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:

fp(x)=gp(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:

hp(x)=fp(x)gp(x).

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

If f is completely multiplicative, then formally:

fp(x)=11−f(p)x.

## Examples

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

* The Möbius function μ has μp(x)=1−x.
* The Mobius function squared has μp2(x)=1+x.
* Euler's totient φ has φp(x)=1−x1−px.
* The multiplicative identity of the Dirichlet convolution δ has δp(x)=1.
* The Liouville function λ has λp(x)=11+x.
* The power function Idk has (Idk)p(x)=11−pkx. Here, Idk is the completely multiplicative function Idk(n)=nk.
* The divisor function σk has (σk)p(x)=1(1−pkx)(1−x).
* The constant function, with value 1, satisfies 1p(x)=(1−x)−1, i.e., is the geometric series.
* If f(n)=2ω(n)=∑d|nμ2(d) is the power of the prime omega function, then fp(x)=1+x1−x.
* Suppose that f is multiplicative and g is any arithmetic function satisfying f(pn+1)=f(p)f(pn)−g(p)f(pn−1) for all primes p and n≥1. Then fp(x)=(1−f(p)x+g(p)x2)−1.
* If μk(n)=∑dk|nμk−1(ndk)μk−1(nd) denotes the Möbius function of order k, then (μk)p(x)=1−2xk+xk+11−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

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