On the intervals between consecutive numbers that are sums of two primes
type: Item
instance of: scholarly article
title: On the intervals between consecutive numbers that are sums of two primes
zbMATH Open document ID: 0798.11040
author: H. Mikawa
published in: Tsukuba Journal of Mathematics
publication date: 1994-07-21T00:00:00Z
review text: Let \(g_ 1<g_ 2 < \cdots\) be the sequence of Goldbach numbers, that is to say the even numbers that are the sum of two primes. It follows from the results of \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)] that \(g_{n + 1}-g_ n = 2\) for all \(g_ n \leq x\), with \(O(x^{1-\delta})\) exceptions, and that \(g_{n + 1} -g_ n \ll g_ n^{7/72 + \varepsilon}\) in all cases. The present paper uses an ingenious method, based on zero-density estimates for the Riemann zeta-function, to show that \[ \sum_{g_ n \leq x} (g_{n + 1} - g_ n)^ 3 \ll x(\log x)^{300}, \] and hence that \[ \sum_{g_ n \leq x} (g_{n + 1} - g_ n)^ \gamma \sim 2^{\gamma-1}x \] for \(0 \leq \gamma<3\).
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 591455
zbMATH Keywords: Goldbach problem
zbMATH Keywords: short intervals
zbMATH Keywords: differences
zbMATH Keywords: Goldbach numbers
zbMATH Keywords: zero- density estimates
zbMATH Keywords: Riemann zeta-function
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162271
OpenAlex ID: W1899442861
DOI: 10.21099/TKBJM/1496162271
description: scientific article; zbMATH DE number 591455