A class of sums with unexpectedly high cancellation

Abstract: A class of sums with unexpectedly high cancellation Abstract: In this talk we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums, of which $\sum_{n^2\leq x}(-1)^ne^{\sqrt{x-n^2}}$ is an example (to get the idea of the result). It turns out that sums in the class we consider are much smaller than would be predicted by certain probabilistic heuristics. After stating the motivation, we show a number of results in integer partitions. For instance we show a ``weak" version of pentagonal number theorem $$ \sum_{\ell^2 < x} (-1)^\ell p(x-\ell^2)\ \sim\ 2^{-3/4} x^{-1/4} \sqrt{p(x)}, $$ where $p(x)$ is the usual partition function.

Joint work with Ernie Croot.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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The conference program, list of speakers, and abstracts are posted on the external website.

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