# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

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

### Tom 160 / 2013

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

#### Streszczenie

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.

#### Autorzy

• Daniel Alan GoldstonDepartment of Mathematics
San Jose State University
San Jose, CA 95192, U.S.A.
e-mail
• János PintzRényi Mathematical Institute
of the Hungarian Academy of Sciences
H-1364 Budapest, Pf 127, Hungary
e-mail
• Cem Yalçın YıldırımDepartment of Mathematics
Boğaziçi University
Bebek, İstanbul 34342, Turkey
e-mail

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