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Small gaps between primes. (English) Zbl 1306.11073

The prime \(k\)-tuples and small gaps between prime numbers are considered. Using a refinement of the Goldston-Pintz-Yildirim sieve method [D. A. Goldston et al., Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)] the author proves, for instance, the following estimates \[ \liminf_{n\to\infty}\,(p_{n+m}-p_n)\ll m^3\text{{e}}^{4m}, \quad \liminf_{n\to\infty}\,(p_{n+1}-p_n)\leq 600 \] with an absolute constant in sign \(\ll\). Here \(m\) is a natural number, and \(p_{\,l}\) denote the \(l\)-th prime number.

MSC:

11N05 Distribution of primes
11N36 Applications of sieve methods

Citations:

Zbl 1207.11096
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References:

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