On the intervals between consecutive numbers that are sums of two primes (Q1327644)

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 \(0 \leq \gamma<3\).