Differences between consecutive primes. (English) Zbl 0639.10027

Heath-Brown, D. R.

Jahresber. Dtsch. Math.-Ver. 90, No. 2, 71-89 (1988).

This is a valuable and comprehensive survey on what is known about the differences between consecutive primes. The author, who has made important contributions to this subject, outlines the main conjectures on these differences together with numerical evidence and heuristic arguments supporting these conjectures. The paper is divided into seven sections with the following titles: 1. Introduction - a Probabilistic Model. 2. The Conjectures - Small Gaps and Large Gaps. 3. Small Gaps. 4. Lower Bounds on Large Gaps. 5. The Riemann Zeta-function and the Prime Number Theorem. 6. Upper Bounds for \(p_{n+1}-p_ n\) via Zero-Density Estimates. 7. Upper Bounds for \(p_{n+1}-p_ n\) Using Sieve Methods. The last section describes an approach to the problem which has not yet been published in full.

Reviewer: T.M.Apostol

MSC:

11N05	Distribution of primes
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11N35	Sieves

Keywords:

survey; differences between consecutive primes; Small Gaps; Large Gaps; Lower Bounds; Riemann Zeta-function; Prime Number Theorem; Upper Bounds; Zero-Density Estimates; Sieve Methods