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Arithmetic progressions of prime-almost-prime twins. (English) Zbl 0929.11043

We say that a positive integer \(n\) is \(P_r\) if its total number of prime factors does not exceed \(r\), that is if \(\Omega(n)\leq r\). Following, J. Chen [Sci. Sinica 16, 157-176 (1973; Zbl 0319.10056)] who proved that there are infinitely many primes \(p\) such that \(p+2\) is \(P_2\), and D. R. Heath-Brown [J. Lond. Math. Soc. (2) 23, 396-414 (1981; Zbl 0448.10046)] who proved that there are infinitely many four-term arithmetic progressions whose first three terms are primes and the fourth is \(P_2\), the author studies three-term arithmetic progressions of distinct primes \(p_1\), \(p_2\) and \(p_3=\frac 12(p_1+p_2)\), showing that there are infinitely many such sequences satisfying \(p_1+2=P_5\), \(p_2+2=P_5\) and \(p_3+2=P_8\). The same technique also yields a variety of similar results. T. P. Peneva and the author [Acta Arith. 83, 155-169 (1998; Zbl 0898.11037)] had previously proved that there exist infinitely many triples as above with \((p_1+2)(p_2+2)=P_9\).
The proof is a combination of the circle method and Iwaniec’s vector sieve [H. Iwaniec, Journées de théorie additive des nombres, Université de Bordeaux I, 71-89 (1977; Zbl 0521.10016)]: its turning point is an upper bound for a suitable weighted average of the generating function for this problem over the minor arcs. The author himself points out that it would be possible to prove that \(p_3+2\) is also \(P_5\) provided that in this key estimate one could raise the level of distribution from essentially \(x^{1/3}\) to \(x^{1/2}\).

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11N36 Applications of sieve methods
11P55 Applications of the Hardy-Littlewood method
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