## Primes in tuples IV: Density of small gaps between consecutive primes

### Volume 160 / 2013

Acta Arithmetica 160 (2013), 37-53
MSC: Primary 11N05; Secondary 11N36.
DOI: 10.4064/aa160-1-3

#### Abstract

We prove that given any small but fixed $\eta > 0$, a positive proportion of all gaps between consecutive primes are smaller than $\eta $ times the average gap. We show some unconditional and conditional quantitative results in this vein. In the results the dependence on $\eta $ 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.