## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  023.29801
Autor:  Erdös, Pál
Title:  The difference of consecutive primes. (In English)
Source:  Duke Math. J. 6, 438-441 (1940).
Review:  Let pn denote the n-th prime, and let A = liminf {pn+1-pn \over log n}. Hardy and Littlewood proved a few years ago, by using the Riemann Hypothesis (= R.H.) that A \leq 2/3 (not yet published), and R.A.Rankin [Proc. Cambridge Philos. Soc. 36, 255-266 (1940; Zbl 025.30702)] recently proved, again by using R. H. that A \leq 3/5 . Depending on Brun's method the author proves without R. H. that A < 1-c for a certain c > 0. Denote by q1 < q2 ··· < qy the primes not exceeding n. Then the author enunciates the following conjecture: sumi = 1y-1 (qi+1-qi)2 = O(n log n). This is, if true, a stronger result than that of H.Cramér [Acta Arith. 2, 23-46 (1936; Zbl 015.19702)].
Reviewer:  S.Ikehara (Osaka)
Classif.:  * 11N05 Distribution of primes
Index Words:  Number theory

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