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Small differences between consecutive primes

  • M. N. Huxley (a1)

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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]

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References

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1. Bombieri, E. and Davenport, H., “Small differences between prime numbers”, Proc. Royal Soc. A293 (1966), 118.
2. Huxley, M. N., “On the differences of primes in arithmetical progressions”, Acta Arithmetica, 15 (1969), 367392.
3. Pilt'ai, G. Z., “On the size of the difference between consecutive primes”, Issledovania po teorii chisel, 4 (1972), 7379.
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