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Differences between consecutive primes. (English) Zbl 0891.11046
Let $$p_n$$ be the $$n$$-th prime. This paper is concerned with proving the following result on the distribution of consecutive primes. Theorem. $\sum_{p_{n+1}-p_n> x^{\frac12}, p_n\leq x}p_{n+1}-p_n\;\ll\;x^{\frac{25}{36}+ \varepsilon}. \tag{1}$ The exponent of $$x$$ in this theorem improves on the work of Heath-Brown who proved (1) with exponent $$\frac34$$. On the Riemann hypothesis one can prove (1) with exponent $$\frac12$$. The proof of the theorem starts off with the Heath-Brown–Linnik identity, which leads to a formula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. The author then estimates the coefficients by using among other things the information which can be gained from Montgomery’s mean value theorem and Huxley’s version of the Halász lemma. Furthermore by using familiar sieve arguments he is able to discard some of the coefficients, allowing an improvement over the previous result of Heath-Brown.
Reviewer: A.Peck (Oxford)

##### MSC:
 11N05 Distribution of primes 11N36 Applications of sieve methods
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