The Bell Series of an Arithmetical Function

Given an arithmetical functionand a primewe denote bythe power seriesand call this the Bell series ofmodulo

Bell series are very useful whenis multiplicative. We need the following theorem.

Theorem

The Bell series for an arithmetical function is unique. Supposeandare multiplicative functions, thenif and only iffor all primes

Proof: Ifthenfor alland allsoConversely iffor allthenfor allSinceandare multiplicative and take the same values for all prime powers they agree for all the positive integers so

Examples

Bell series for the Mobius function. Sinceandforwe have

Bell series for Euler's totient functionSinceforwe have

Ifis a completely multiplicative function so thatfor allthen the Bell series is a geometric series and

as long as

Hence we have the following