Number Theory Seminar
Wednesday, February 13, 2019, 3:15pm, 119 Physics
Lillian Pierce (Duke University)
Short character sums
Abstract:- A surprisingly diverse array of problems in analytic number
theory have at their heart a problem of bounding an
exponential sum, or its multiplicative cousin, a character
sum. In general, the sums of
interest fall into one of two main regimes: complete sums or
incomplete sums, with this latter regime including in particular
“short sums.” Short sums are particularly useful, and particularly
resistant to almost all known methods. In this talk, we will see what
makes a sum “short,” sketch why it would be incredibly powerful to
understand short sums, and discuss a curious proof from the 1950’s
which is still the best way we know to bound short sums. We will end
by describing new work on bounding short sums in much more general situations.
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