Longer than average intervals containing no primes

Cheer, A. Y.; Goldston, D. A.

doi:10.1090/S0002-9947-1987-0911080-7

Longer than average intervals containing no primes
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by A. Y. Cheer and D. A. Goldston

Trans. Amer. Math. Soc. 304 (1987), 469-486

DOI: https://doi.org/10.1090/S0002-9947-1987-0911080-7

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Abstract:

We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first $n$ moments are Poisson. The related problem of longer than average gaps between primes is also considered.

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