### The Bounded Gap Conjecture and bounds between consecutive Goldbach numbers

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At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :

Mertens’ product formula asserts that $$\prod _{p\le x}\left(1-\frac{1}{p}\right)\phantom{\rule{0.166667em}{0ex}}logx\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{e}^{-\gamma}$$ as $x\to \infty $. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^{8}$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

We prove that given any small but fixed η > 0, a positive proportion of all gaps between consecutive primes are smaller than η times the average gap. We show some unconditional and conditional quantitative results in this vein. In the results the dependence on η is given explicitly, providing a new quantitative way, in addition to that of the first paper in this series, of measuring the effect of the knowledge on the level of distribution of primes.

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