Some singular series averages and the distribution of Goldbach numbers in short intervals (Q1342702)

An even number is called a Goldbach number if it can be written as the sum of two primes. It was shown independently by \textit{I. Kátai} [Magyar Tud. Akad. Mat. Fiz. Tud. Oszt. Közl. 17, 99-100 (1967; Zbl 0145.049)] and \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353- 370 (1975; Zbl 0301.10043)] that, under the assumption of the generalized Riemann hypothesis, every interval \((N,N + C \ln^ 2 N]\) (\(C\) an absolute positive constant) contains a Goldbach number. Following ideas of the second author [Glasg. Math. J. 32, 285-297 (1990; Zbl 0719.11065), Mathematika 39, 10-17 (1992; Zbl 0758.11037)] the authors prove various hypothetical results which support the hope that even much shorter intervals contain Goldbach numbers. The unproved hypotheses are: \[ \int^ N_ 1 (\Psi(x + h) - \Psi(x) - h)^ 2 dx = h N \log (N/h) + O(h N)\tag\text{i} \] (\(1\leq h \leq \log N\), \(\Psi(y) = \sum_{n \leq y} \Lambda(n)\)), (ii) a strong form of the Bombieri-Vinogradov mean value theorem (with \(Q = N \text{exp}(-(\log N)^{2/3})\) or even larger). Then the length of the interval can be reduced to \((\log N)^{{2/3} + \varepsilon}\) (or even \((\log \log N)^ B\) for some \(B\)).