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Proceedings of the American Mathematical Society

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Gaps between prime numbers

Authors: Adolf Hildebrand and Helmut Maier
Journal: Proc. Amer. Math. Soc. 104 (1988), 1-9
MSC: Primary 11N05
MathSciNet review: 958032
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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 [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1988 American Mathematical Society