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Following attempts at an analytic proof of the Pentagonal Number Theorem, wereport on the discovery of a general principle leading to an unexpectedcancellation of oscillating sums. After stating the motivation, and ourtheorem, we apply it to prove several results on the Prouhet-Tarry-EscottProblem, integer partitions, and the distribution of prime numbers. Regardingthe Prouhet-Tarry-Escott problem, we show that \begin{align*} \sum_{|\ell|\leqx}(4x^2-4\ell^2)^{2r}-\sum_{|\ell|<x}(4x^2-(2\ell+1)^2)^{2r}=\text{polynomialw.r.t. } x \text{ with degree }2r-1. \end{align*} This can perhaps be provedusing properties of Bernoulli polynomials, but the claim fell out of our methodin a more natural and motivated way. Using this result, we solve an approximateversion of the PTE Problem, and in doing so our work in the approximate caseexceeds the bounds one can prove using a pigeonhole argument, which seemsremarkable. Also, we prove that $$ \sum_{\ell^2 < n} (-1)^\ell p(n-\ell^2)\\sim\ (-1)^n 2^{-3/4} n^{-1/4} \sqrt{p(n)}, $$ where $p(n)$ is the usualpartition function. We get the following "Weak pentagonal number theorem", inwhich we can replace the partition function $p(n)$ with Chebyshev $\Psi$function: $$ \sum_{0 < \ell < \sqrt{xT}/2} \Psi([e^{\sqrt{x -\frac{(2\ell)^2}{T}}},\ e^{\sqrt{x - \frac{(2\ell-1)^2}{T}}}])\=\Psi(e^{\sqrt{x}})\left(\frac{1}{2} + O\left(e^{-0.196\sqrt{x}}\right)\right), $$ where $T=e^{0.786\sqrt{x}}$, where$\Psi([a,b]) := \sum_{n\in [a,b]} \Lambda(n)$ and $\Psi(x) = \Psi([1,x])$,where $\Lambda$ is the von Mangoldt function. Note that this last equation (sumover $\ell$) is stronger than one would get using a strong form of the PrimeNumber Theorem and also a naive use of the Riemann Hypothesis in each interval,since the widths of the intervals are smaller than $e^{\frac{1}{2} \sqrt{x}}$,making the RH estimate ``trivial".

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