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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—Edward T. Hall (b. 1914)