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 $mu$ has $mu\_p(x)=1-x.$
• 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 $(textrm\{Id\}\_k)\_p(x)=frac\{1\}\{1-p^kx\}.$ Here, Id_k is the completely multiplicative function $operatorname\{Id\}\_k(n)=n^k$.
• The divisor function $sigma\_k$ has $(sigma\_k)\_p(x)=frac\{1\}\{1-sigma\_k(p)x+p^kx^2\}.$

References