On limit points of the sequence of normalized prime gaps

Abstract

Let p n denote the nth smallest prime number, and let L denote the set of limit points of the sequence tpp n`1´pn q{ log p n u 8 n"1 of normalized differences between consecutive primes. We show that for k " 9 and for any sequence of k nonnegative real numbers β 1 ď β 2 﨨¨ď β k , at least one of the numbers β j´βi (1 ď i ă j ď k) belongs to L. It follows that at least 12.5% of all nonnegative real numbers belong to L.

The twin prime conjecture

“…Although new ideas appear to be required to prove the Twin Prime Conjecture, the breakthroughs of Theorems 7, 8 and 9 have already had several further applications, including new results on large gaps between primes [21,20,52], the resolution of the Erdős discrepancy problem [65], as well as many other results on the distribution of primes [67,51,69,12,6,28,57,42,2,49,3,16,72,70,55,56,33,4,5,1,37,53,58,1,45,43] and correlations of multiplicative functions [68,48,39,38,30,41,27,26].…”

Gaps Between Primes

“…The fact that one can restrict the entire argument to an arithmetic progression also allows one to get some control on the joint distribution of various arithmetic functions. There have been many recent works making use of these flexibilities in the setup of the sieve method, including [58,13,7,21,48,34,3,39,4,14,61,59,46,47,28,5,6,1,32,43,49].…”

On a Conjecture of Erdős, Pólya and Turán on Consecutive Gaps Between Primes

“…The proof will be based on the first assertion (see (i)) of a very nice result of W. D. Banks, T. Freiberg and J. Maynard [1] which appears as Theorem 4.3 in their work, which we quote now restricted on (i) and with a slight change as…”

“…If k is a sufficiently large multiple of 16m+1 and ǫ is sufficiently small, there is some N (m, k, ǫ) such that the following holds for all N N (m, k, ǫ). With Z N 4ǫ given by (4.8) of [1], let…”

“…Remark 1. The definition of Z N 4ǫ is given earlier in the work [1] but its value does not play a significant role in the application of the result (it is the greatest prime factor of a possible exceptional modulus if such a modulus exists and it is equal to 1 if no such modulus exists).…”