In mathematics, the Bell series is a formal power series used to study properties of multiplicative 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

$f\_p(x)=\backslash sum\_\{n=0\}^\backslash infty\; f(p^n)x^n.$

Two series 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

$h\_p(x)=f\_p(x)\; g\_p(x).\backslash ,$

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

If $f$ is completely multiplicative, then

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

Examples

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

• The Moebius function $\backslash mu$ has $\backslash mu\_p(x)=1-x.$
• Euler's Totient $\backslash phi$ has $\backslash phi\_p(x)=\backslash frac\{1-x\}\{1-px\}.$
• The identity function $I$ has $I\_p(x)=1.$
• The Liouville function $\backslash lambda$ has $\backslash lambda\_p(x)=\backslash frac\{1\}\{1+x\}.$
• The power function Id_k has $(\backslash textrm\{Id\}\_k)\_p(x)=\backslash frac\{1\}\{1-p^kx\}.$ Here, Id_k is the completely multiplicative function $\backslash operatorname\{Id\}\_k(n)=n^k$.
• The divisor function $\backslash sigma\_k$ has $(\backslash sigma\_k)\_p(x)=\backslash frac\{1\}\{1-\backslash sigma\_k(p)x+p^kx^2\}.$

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639

Arithmetic functions | Mathematical series

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