Bell series

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:

Examples

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

  • The Möbius function has
  • The Mobius function squared 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
  • The unit function satisfies , i.e., is the geometric series.
  • If is the power of the prime omega function, then
  • Suppose that f is multiplicative and g is any arithmetic function satisfying for all primes p and . Then
  • If denotes the Mobius function of order k, then
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See also

  • Bell numbers

References

  • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
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