Generating Functions
The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:
The Lambert series generating function is
which converges for |q| < 1.
Both of these are proved by elementary series manipulations and the formulae for φ(n).
Read more about this topic: Euler's Totient Function
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“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)