Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 1 / d / Solution

Every nonprincipal Dirichlet character modulo the prime $q$ is primitive, so its finite Fourier coefficients have modulus one off zero; the coefficient at zero is zero by character orthogonality. Fourier inversion theorem givesThe finite geometric series givesThe printed hint omits $n$ from the exponential; its literal constant summand would not obey the bound for arbitrary $N$. The geometric-series calculation proves the needed estimate independently. Pairing $a$ with $q-a$ givesThe last sum is a harmonic number. This proves the Pólya–Vinogradov inequality uniformly in $M$ and $N$; complete blocks of length $q$ also vanish by Orthogonality of Dirichlet characters.