On twisted character sums

In this paper we give a simple proof of a result by Burgess about short sums involving Dirichlet characters and exponentials. Indeed we establish a slightly stronger and more general bound that applies to sums of the form ({sum_{n=M+1}^{M+N}f(alpha n)chi(n)}), where χ is a non-principal character to the modulus p and f is a smooth 1-periodic function.