Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

@article{Granville2005LargeCS,
  title={Large character sums: Pretentious characters and the P{\'o}lya-Vinogradov theorem},
  author={Andrew Granville and Kannan Soundararajan},
  journal={Journal of the American Mathematical Society},
  year={2005},
  volume={20},
  pages={357-384}
}
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for… 

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TLDR
This paper proves a lower bound for maxχ≠χ0|∑n⩽xχ(n) so that the expected maximal value of character sums for most characters is recovered.
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