Small Gaps between Primes Ii (preliminary)

We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞ (pn+1 − pn) log pn(log log pn)−1 log log log log pn < ∞. Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ X with any level θ > 1/2 we have bounded differences between consecutive primes infinitely often: (1.2) lim inf n→∞ (pn+1 − pn) ≤ C(θ) with a constant C(θ) depending only on θ. If the Bombieri–Vinogradov theorem holds with a level θ > 20/21, in particular if the Elliott–Halberstam conjecture holds, then we obtain (1.3) lim inf n→∞ (pn+1 − pn) ≤ 20, that is pn+1 − pn ≤ 20 for infinitely many n. Inequalities (1.2)–(1.3) will follow from the even stronger following result Theorem A. Suppose the Bombieri–Vinogradov theorem is true for Q ≤ X with some θ > 1/2. Then there exists a constant C ′(θ) such that any admissible k-tuple contains at least two primes for any (1.4) k ≥ C ′(θ) if θ > 1/2, where C ′(θ) is an explicitly calculable constant depending only on θ. Further we have at least two primes for (1.5) k = 7 if θ > 20/21. Remark. For the definition of admissibility see (2.2) below. We will show some more general results for the quantity (ν is a given positive integer) (1.6) Eν = lim inf n→∞ pn+ν − pn log pn . Date: February 8, 2005. 1991 Mathematics Subject Classification. Primary 11N05; Secondary 11P32.