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- W2976295524 abstract "Following attempts at an analytic proof of the Number Theorem, we report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums, of which $sum_{n^2leq x}(-1)^ne^{sqrt{x-n^2}}$ 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. Regarding the Prouhet-Tarry-Escott problem, we show that begin{align*} sum_{|ell|leq x}(4x^2-4ell^2)^{2r}-sum_{|ell|<x}(4x^2-(2ell+1)^2)^{2r}=text{polynomial w.r.t. } x text{ with degree }2r-1. end{align*} Using this result, we solve an approximate version of the Prouhet-Tarry-Escott Problem, and in doing so we give some evidence that a certain pigeonhole argument for solving the exact version of the Problem can be improved. Also we prove $$ sum_{ell^2 < x} (-1)^ell p(x-ell^2) sim 2^{-3/4} x^{-1/4} sqrt{p(x)}, $$ where $p(x)$ is the usual partition function; and also prove the following Pentagonal Number Theorem for the Primes, which counts the number of primes (with von Mangoldt weight) in a set of intervals very precisely: $$ sum_{0 leq 2ell < sqrt{xT}} Psi([e^{sqrt{x - (2ell)^2/T}}, e^{sqrt{x - (2ell-1)^2/T}}]) {Psi(e^{sqrt{x}}) over 2} Oleft ( e^{sqrt{x}} T^{-frac{1}{4} + o(1)} right ), $$ where $T = e^{4alpha sqrt{x}/3}$, where $alpha = 1 - sqrt{frac{2}{2+pi^2}}$, where $Psi([a,b]) := sum_{nin [a,b]} Lambda(n)$ and $Psi(x) = Psi([1,x])$, where $Lambda$ is the von Mangoldt function. Note that this last equation (sum over $ell$) is stronger than one would get using a strong form of the Prime Number Theorem and also the Riemann Hypothesis, since the widths of the intervals are smaller than $e^{frac{1}{2} sqrt{x}}$, making the Riemann Hypothesis estimate trivial." @default.
- W2976295524 created "2019-10-03" @default.
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- W2976295524 date "2019-09-27" @default.
- W2976295524 modified "2023-09-27" @default.
- W2976295524 title "On a Class of Sums with Unexpectedly High Cancellation, and its Applications" @default.
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