- Bell series
In
mathematics , the Bell series is aformal power series used to study properties of arithmetical functions. Bell series were introduced and developed byEric Temple Bell .Given an
arithmetic function and a prime , define the formal power series , called the Bell series of modulo as:
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 function s and , one hasif and only if : for all primes .Two series may be multiplied (sometimes called the "multiplication theorem"): For any two
arithmetic function s and , let be theirDirichlet convolution . Then for every prime , one has:
In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.
If is
completely multiplicative , then :Examples
The following is a table of the Bell series of well-known arithmetic functions.
* The
Moebius function has
* Euler's Totient has
* The multiplicative identity of theDirichlet convolution has
* TheLiouville function has
* The power function Idk has Here, Idk is the completely multiplicative function .
* Thedivisor function hasReferences
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