Chains of large gaps between consecutive primes

Abstract

Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) :::: (c + o(I)) logx loglogx log log log 10gx(loglog logx)-2, where c = e Y and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4/ Co-e-4/co = 3. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.

Gaps between prime numbers

“…The proof of the theorem follows closely the argument of [5]. In this section we state four lemmas, all of which have their counterparts in [5].…”

“…In [5] the second author extended this result to an arbitrary number of consecutive gaps, showing that for any k > 1…”

“…The purpose of this note is to prove the following theorem, which settles Erdös' question in the affirmative. Roughly speaking, the theorem asserts that a positive proportion of all real numbers belong to S, and that an analogous result holds for the limit points (in Rfc) of the sequence (5) fA.,. "..v*gii} (» = 1 The theorem immediately implies that the sequence {ein/logn} has arbitrarily large finite limit points, thus answering the above-mentioned question of Erdös.…”

“…In this section we state four lemmas, all of which have their counterparts in [5]. We shall give a detailed proof only for the last lemma; the first three lemmas are obtained by minor modifications of the proofs in [5].…”

“…For the proof of the theorem we shall use a method that was introduced by the second author in [5] to prove (2). The key idea is to construct a matrix, whose rows are intervals of consecutive integers, and which contains exceptionally few primes.…”