Abstract
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.
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