Limit points and long gaps between primes

2015/10/27 by Roger C. Baker, Baker, Roger, Tristan Freiberg +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Finite Group Theory Research

paper · doi:10.48550/arxiv.1510.08054

Abstract

Let d_n = p_n+1 - p_n, where p_n denotes the nth smallest prime, and let R(T) = log T log₂ Tlog₄ T/(log₃ T)² (the "Erd\H os--Rankin" function). We consider the sequence (d_n/R(p_n)) of normalized prime gaps, and show that its limit point set contains at least 25% of nonnegative real numbers. We also show that the same result holds if R(T) is replaced by any "reasonable" function that tends to infinity more slowly than R(T)log₃ T. We also consider "chains" of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between primes with subsequent developments of Ford, Green, Konyagin, Maynard and Tao on long gaps between consecutive primes.

Citations

• On the distribution of gaps between consecutive primes

Related

• A note on the distribution of normalized prime gaps
• On the distribution of gaps between consecutive primes
• On the ratio of consecutive gaps between primes
• Rough numbers between consecutive primes
• Large gaps between primes