eulers phi-function

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 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



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

  • The Moebius function mu has mu_p(x)=1-x.
  • Euler's Totient phi has phi_p(x)=frac{1-x}{1-px}.
  • The multiplicative identity of the Dirichlet convolution delta has delta_p(x)=1.
  • The Liouville function lambda has lambda_p(x)=frac{1}{1+x}.
  • The power function Idk has (textrm{Id}_k)_p(x)=frac{1}{1-p^kx}. Here, Idk is the completely multiplicative function operatorname{Id}_k(n)=n^k.
  • The divisor function sigma_k has (sigma_k)_p(x)=frac{1}{1-sigma_k(p)x+p^kx^2}.


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