Gaps between prime numbers

Hildebrand, Adolf; Maier, Helmut

doi:10.1090/S0002-9939-1988-0958032-5

Gaps between prime numbers
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by Adolf Hildebrand and Helmut Maier PDF

Proc. Amer. Math. Soc. 104 (1988), 1-9 Request permission

Abstract:

Let ${d_n} = {p_{n + 1}} - {p_n}$ denote the $n$th gap in the sequence of primes. We show that for every fixed integer $k$ and sufficiently large $T$ the set of limit points of the sequence $\{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\}$ in the cube ${[0,T]^k}$ has Lebesgue measure $\geq c(k){T^k}$, where $c(k)$ is a positive constant depending only on $k$. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence $\{ {d_n}/\log n\}$ has a finite limit point greater than 1.

References

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Helmut Maier, Chains of large gaps between consecutive primes, Adv. in Math. 39 (1981), no. 3, 257–269. MR 614163, DOI 10.1016/0001-8708(81)90003-7

Small differences between prime numbers

The difference between consecutive prime numbers

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Giovanni Ricci, Recherches sur l’allure de la suite $\{p_{n+1}-p_n/\log p_n\}$, Colloque sur la Théorie des Nombres, Bruxelles, 1955, Georges Thone, Liège; Masson & Cie, Paris, 1956, pp. 93–106 (French). MR 0079605

Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind

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