On the distribution of gaps between consecutive primes

Janos Pintz

Published 2014-07-08, updated 2014-09-24Version 2

Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an interval of type [0,c] with a positive ineffective value c. In the present work we extend this result for a large class of normalizing functions. The only essential requirement is that the function f(n) replacing logn should satisfy f(n)<<lognloglognloglogloglogn/(logloglogn)^2 (with a small implied constant), the well-known Erd\"os-Rankin bound for the largest known gaps between consecutive primes. The work also proves that apart from a thin set of exceptional functions the original Erd\"os conjecture holds if logn is replaced by a non-exceptional function f(n). The paper also gives a new proof for a result of Helmut Maier which generalized the Erd\"os-Rankin bound for an arbitrarily long finite chain of consecutive primegaps. The proof uses a combination of methods of Erd\"os-Rankin,Maynard-Tao and Banks-Freiberg-Maynard. Since the submission of the present work the very important recent simultaneous and independent works of Ford-Green-Konjagin-Tao (arXiv:1408.4505 [math.NT] and Maynard (aerXiv:1408.5110 [math.NT]) appeared on arXiv and they proved the old conjecture of Erd\"os which asserts that the lower bound for large gaps exceeds Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. In this new version we prove the same assertions as in the original work for the case when f(n)<<Clognloglognloglogloglogn/(logloglogn)^2 with an arbi8trarily large constant C, in particular we show that there are blocks of m primes for any m such that all gaps between these primes simultaneously satisfy the lower estimate Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. The proof uses the method of Maynard.