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Primes in tuples. III: On the difference $$p_{n+\nu}-p_n$$. (English) Zbl 1196.11123
Summary: In the present work we prove a new estimate for $$\Delta_\nu:=\liminf_{n \to \infty} \frac{(p_{n+\nu}-p_n)}{\log p_n}$$, where $$p_n$$ denotes the $$n$$th prime. Combining our recent method which led to $$\Delta_1=0$$ with H. Maier’s matrix method [Mich. Math. J. 35, No. 3, 323–344 (1988; Zbl 0671.10037)], we show that $$\Delta_\nu\leq e^{-\gamma}(\sqrt{\nu}-1)^2$$. We also extend the result to primes in arithmetic progressions where the modulus can tend slowly to infinity as a function of $$p_n$$.
In Part I [Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)], the authors found a variation of Selberg’s $$\lambda^2$$-sieve which can be used to prove that $$\Delta_ 1=0$$.
For Part II, see the authors, Acta Math. 204, No. 1, 1–47 (2010; Zbl 1207.11097).

##### MSC:
 11N05 Distribution of primes 11N13 Primes in congruence classes
##### Keywords:
Primes in tuples; primes in arithmetic progressions
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##### References:
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