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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This work was completed under the support of the grants MTM2008-03880/MTM (Ministerio de Ciencia e Innovación) and CCG08-UAM/ESP-3906 (UAM-Comunidad de Madrid).
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Chamizo, F. On twisted character sums. Arch. Math. 96, 417–421 (2011). https://doi.org/10.1007/s00013-011-0255-0
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DOI: https://doi.org/10.1007/s00013-011-0255-0