On a Class of Sums with Unexpectedly High Cancellation, and its Applications

21 Dec 2019 Croot Ernie Mousavi Hamed

Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to the 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). After stating the motivation, and our theorem, we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem... (read more)

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