Residue for the generating function of the Euler totient function

F. Carlson proved in 1921 that a power series with integer coefficients and radius of convergence 1 is either a rational function or has the unit circle as a natural boundary. Since $f(z)$ is clearly not rational, it has the unit circle as a natural boundary. For some references, see the solution to Exercise 4.46(c) of Enumerative Combinatorics, vol. 1, 2nd ed.

Note that $f(z)=\sum_{n=1}^\infty \phi(n) z^n$ can be written as a Lambert series, $$f(z) = \sum a_n\frac{z^n}{1-z^n},$$ where $a_n = \sum_{d|n}\phi(d) \mu(n/d).$ It is clear that the Lambert series converges inside the unit disk, and $f(z)$ has a natural boundary on the unit circle (since it will have a pole at every root of unity).