Short intervals with a given number of primes

Abstract

A. A well-known conjecture asserts that, for any given positive real number λ and nonnegative integer m, the proportion of positive integers n ď x for which the interval pn, n`λ log ns contains exactly m primes is asymptotically equal to λ m e´λ{m! as x tends to infinity. We show that the number of such n is at least x 1´op1q .

Almost primes in almost all short intervals

“…where the notations are as in Subsection 1.2, and w = exp log X (log log X) 3 and R = 2 (log log X) 3 2 .…”

Short intervals containing a prescribed number of primes

“…This is precisely [1, Lemma 3.2], adapted to our situation. In particular, the main difference is in the choice of g. Indeed, here we chose g ∈ [log x, 2 log x] whereas in [1] was chosen g = p≤log x η p. Anyway, the structure of the proof of the Theorem 2.2 is exactly the same of [1, Lemma 3.2] and the computations change only in a few points, in which it is very easy to verify that they do not affect the stated results (2.7) and (2.8). For the aforementioned reasons, we omit the details of the proof.…”

“…for every λ and m. We refer the reader to the expository article [5] of Soundararajan for further discussions on these fascinating statistics. In order to prove Theorem 1.2 we use a combination of ideas present in [1] and [3]. The Freiberg's approach was to construct a special admissible set of linear forms, by using an Erdős-Rankin type construction [1,Lemma 3.3], to which apply the Maynard result [3, Theorem 3.1], finding many disjoint intervals containing at least m primes.…”

“…In order to prove Theorem 1.2 we use a combination of ideas present in [1] and [3]. The Freiberg's approach was to construct a special admissible set of linear forms, by using an Erdős-Rankin type construction [1,Lemma 3.3], to which apply the Maynard result [3, Theorem 3.1], finding many disjoint intervals containing at least m primes. The conclusion now follows by using a sliding process to detect several different intervals containing exactly m primes.…”

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].…”