Bell series - Alchetron, The Free Social Encyclopedia

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 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 ) = ∑ n = 0 ∞ f ( p n ) x n .

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 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 ) .

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

If f is completely multiplicative, then formally:

f p ( x ) = 1 1 − f ( p ) x .

Examples

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

The Möbius function μ has μ p ( x ) = 1 − x .

Euler's Totient φ has φ p ( x ) = 1 − x 1 − p x .

The multiplicative identity of the Dirichlet convolution δ has δ p ( x ) = 1.

The Liouville function λ has λ p ( x ) = 1 1 + x .

The power function Id_k has ( Id k ) p ( x ) = 1 1 − p k x . Here, Id_k is the completely multiplicative function Id k ⁡ ( n ) = n k .

The divisor function σ k has ( σ k ) p ( x ) = 1 1 − ( 1 + p k ) x + p k x 2 .

References

Bell series Wikipedia

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