## 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

Zbl 1207.11096
Full Text:

### References:

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