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Small differences between consecutive primes

Published online by Cambridge University Press:  26 February 2010

M. N. Huxley
M. N. Huxley
Affiliation:
University College, Cardiff.
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Extract

Let pn denote the n-th prime number, and let

It follows from the prime number theorem that E ≤ 1. Erdős used a sieve argument to show that E < 1, and Rankin and Ricci gave the explicit estimates E ≤ 57/59 and E ≤ 15/16 respectively. A more powerful approach used by Hardy and Little-wood and by Rankin depended on hypotheses about the zeros of L-functions until Bombieri's Theorem was found. With its aid Bombieri and Davenport [1] proved that

A fuller history, with bibliography, is to be found in [1]

Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 229 - 232
Copyright
Copyright © University College London 1973

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References

1. Bombieri, E. and Davenport, H., “Small differences between prime numbers”, Proc. Royal Soc. A293 (1966), 1–18.Google Scholar
2. Huxley, M. N., “On the differences of primes in arithmetical progressions”, Acta Arithmetica, 15 (1969), 367–392.CrossRefGoogle Scholar
3. Pilt'ai, G. Z., “On the size of the difference between consecutive primes”, Issledovania po teorii chisel, 4 (1972), 73–79.Google Scholar