Partial isometry

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:

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

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 $\phi$ 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 $({\sigma }_k{)}_p(x)=\frac{1}{1-(1+p^k)x+p^k{x}^2}.$

See also

• Bell numbers

References

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