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Lillian Pierce : Burgess bounds for short mixed character sums

A celebrated result of Burgess proves nontrivial bounds for short multiplicative character sums. In general, bounds for short character sums have utility in a wide range of problems in number theory, and it would be highly desirable to extend BurgessÂ?s method to apply to more general character sums. This talk presents new work in this direction, joint with Roger Heath-Brown, that proves nontrivial bounds for short mixed character sums in which the additive character is evaluated at a real-valued polynomial. Our approach, via a version of the Burgess method, includes a novel application of the recent results of Wooley on the Vinogradov mean value theorem.

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