The Bounded Gap Conjecture and bounds between consecutive Goldbach numbers
Landau’s problems on primes
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 :
On the exceptional set for the -twin primes problem
Primes in Shart Intervals.
On the comparative theory of primes
Über eine Verallgemeinerung des Tschebyschef-Problems.
Oscillation of Mertens’ product formula
Mertens’ product formula asserts that as . Calculation shows that the right side of the formula exceeds the left side for . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on , this and a complementary inequality might change their sense for sufficiently large values of . We show this to be the case.
Primes in tuples IV: Density of small gaps between consecutive primes
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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