  The discrete Fourier transform of the greatest common divisor is a
multiplicative function, if taken with respect to the same order of the
primitive root of unity, which is a well known fact. As such, the transform can
be expressed in the prime factors of the argument, the explicit form of which
is proven in this paper. Subsequently it is shown how the procedure can be
generalized to the discrete Fourier transform of a function of the greatest
common divisor. From this representation some interesting relations concerning
the Euler totient function and generalizations thereof are established.

Number Theory