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 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)=\sum_{n=0}^\infty 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:



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

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