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The addition of primes and power. (English) Zbl 0860.11061

The authors give a second application of the recently developed vector sieve of the first author and E. Fouvry [J. Reine Angew. Math. 454, 59-96 (1994; Zbl 0809.11060)]. They sift the set of vectors \((n,n +2)\), thus getting improved results for the set of integers \(n\) such that \(n\) and \(n+2\) have large prime factors only. The main ingredients of this high-standard work are the Rosser-Iwaniec sieve [H. Iwaniec, Acta Arith. 37, 307-320 (1980; Zbl 0444.10038)] and mean upper bounds for Kloosterman sums due to J.-M. Deshouillers and H. Iwaniec [Invent. Math. 70, 219-288 (1982; Zbl 0502.10021)].

MSC:

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