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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“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)