Definitions

# 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 $f_p\left(x\right)$, called the Bell series of $f$ modulo $p$ as

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

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

$f_p\left(x\right)=g_p\left(x\right)$ 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

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

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

If $f$ is completely multiplicative, then

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

## Examples

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

• The Moebius function $mu$ has $mu_p\left(x\right)=1-x.$
• Euler's Totient $phi$ has $phi_p\left(x\right)=frac\left\{1-x\right\}\left\{1-px\right\}.$
• The multiplicative identity of the Dirichlet convolution $delta$ has $delta_p\left(x\right)=1.$
• The Liouville function $lambda$ has $lambda_p\left(x\right)=frac\left\{1\right\}\left\{1+x\right\}.$
• The power function Idk has $\left(textrm\left\{Id\right\}_k\right)_p\left(x\right)=frac\left\{1\right\}\left\{1-p^kx\right\}.$ Here, Idk is the completely multiplicative function $operatorname\left\{Id\right\}_k\left(n\right)=n^k$.
• The divisor function $sigma_k$ has $\left(sigma_k\right)_p\left(x\right)=frac\left\{1\right\}\left\{1-sigma_k\left(p\right)x+p^kx^2\right\}.$

## References

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