Bell series

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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 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 functions and , one has if and only if:

for all primes .

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions and , let be their Dirichlet 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 formally:


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

  • The Möbius function has
  • Euler's Totient has
  • The multiplicative identity of the Dirichlet convolution has
  • The Liouville function has
  • The power function Idk has Here, Idk is the completely multiplicative function .
  • The divisor function has

See also[edit]